On Embedding and NP-Complete Problems of Equitable Labelings
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This document discusses embedding and NP-complete problems related to equitable graph labelings. It begins by introducing concepts like cordial labeling, edge product cordial labeling, total edge product cordial labeling, difference cordial labeling, simply sequential labeling, total sequential cordial labeling, and divisor cordial labeling. It then presents proofs that any graph G can be embedded as an induced subgraph of graphs that admit these different labeling schemes, such as total edge product cordial graphs, difference cordial graphs, total sequential cordial graphs, and divisor cordial graphs. The document explores how to construct supergraphs of G that satisfy the conditions for these labeling types by adding vertices or edges to G in certain ways.
![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 80-85
www.iosrjournals.org
DOI: 10.9790/5728-11138085 www.iosrjournals.org 80 |Page
On Embedding and NP-Complete Problems of Equitable
Labelings
S. K. Vaidya1
, C. M. Barasara2
1
(Department of Mathematics, Saurashtra University, Rajkot - 360 005, Gujarat, INDIA)
2
(Atmiya Institute of Technology and Science, Rajkot - 360 005, Gujarat, INDIA)
Abstract: The embedding and NP-complete problems in the context of various graph labeling schemes were
remained in the focus of many researchers. Here we have explored embedding and NP-Complete problems for
some variants of cordial labeling.
Keywords: Graph labeling, equitable labeling, embedding of graphs, NP-complete problems.
Mathematics Subject Classification (2010): 05C78, 05C60.
I. Introduction
A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain
condition or conditions.
Labeling of discrete structures is a potential area of research and a comprehensive survey on graph
labeling can be found in Gallian [1].
Definition 1.1: For a graph G , a vertex labeling function }1,0{)(: GVf induces an edge labeling function
}1,0{)(:*
GEf defined as )()()(*
vfufuvef . Then f is called cordial labeling of graph G if
absolute difference of number of vertices with label 1 and label 0 is at most 1 and absolute difference of number
of edges with label 1 and label 0 is at most 1.
A graph G is called cordial if it admits a cordial labeling.
The concept of cordial labeling was introduced by Cahit [2] in 1987 and in the same paper he presented
several results on this newly introduced concept.
After this some labelings like prime cordial labeling, A - cordial labeling, product cordial labeling,
edge product cordial labeling etc. were also introduced with minor variations in cordial theme. Such labelings
are classified as equitable labelings.
Definition 1.2: For a graph G , an edge labeling function }1,0{)(: GEf induces a vertex labeling function
}1,0{)(:*
GVf defined as )}(/)({)(*
GEuvuvfuf . Then f is called an edge product cordial labeling
of graph G if the absolute difference of number of vertices with label 1 and label 0 is at most 1 and the absolute
difference of number of edges with label 1 and label 0 is also at most 1.
A graph G is called edge product cordial graph if it admits an edge product cordial labeling.
In 2012, the concept of edge product cordial labeling was introduced by Vaidya and Barasara [3]. In
the same paper they have investigated edge product cordial labeling for some standard graphs.
For any graph G , we introduce following notations:
1. )1(fv = the number of vertices having label 1;
2. )0(fv = the number of vertices having label 0;
3. (1)fe = the number of edges having label 1;
4. (0)fe = the number of edges having label 0;
Throughout this discussion we consider simple, finite and undirected graph ))(),(( GEGVG with
pGV )( and qGE )( . For all other standard terminology and notation we refer to Chartrand and
Lesniak[4].](https://image.slidesharecdn.com/n011138085-151123095857-lva1-app6891/85/On-Embedding-and-NP-Complete-Problems-of-Equitable-Labelings-1-320.jpg)
![On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 81 |Page
II. Embedding of Equitable graphs
Definition 2.1: For a graphG , an edge labeling function }1,0{)(: GEf induces a vertex labeling function
}1,0{)(:*
GVf defined as )}(/)({)(*
GEuvuvfuf . Then f is called a total edge product cordial
labeling of graph G if (1) (1) (0) (0) 1f f f fv e v e . A graph G is called total edge product cordial
graph if it admits a total edge product cordial labeling.
The concept of total edge product cordial labeling was introduced by Vaidya and Barasara [5].
Theorem 2.2: Any graph G can be embedded as an induced subgraph of a total edge product cordial graph.
Proof. Label the edges of G in such a way that edges with label 1 and edges with label 0 differ by at most 1.
Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and
in E be the cardinality of set iV and iE respectively.
Case 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 0 to these new edges. Consequently
2
r
vertices will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Case 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 1 to these new edges. Consequently
2
r
vertices will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total edge
product cordial graph. That is, any graph G can be embedded as an induced subgraph of a total edge product
cordial graph.
Definition 2.3: Let : ( ) {1,2,..., }f V G p be a bijection. For each edge uv , assign the label ( ) ( )f u f v .
Function f is called a difference cordial labeling if (1) (0) 1f fe e . where (1)fe and (0)fe denote the
number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is
called a difference cordial graph.
The concept of difference cordial labeling was introduced by Ponraj et al. [6] in 2003.
Theorem 2.4: Any graph G can be embedded as an induced subgraph of a difference cordial graph.
Proof. Let G be the given graph with p vertices and q edges. As per definition of difference cordial labeling
assign labels {1,2,..., }p to the vertices of G .
Let iE be the set of edges with label i while in E be the cardinality of set iE .
Case 1: When 1 0( ) ( ) 1n E n E (say r ).
The new graph H can be obtained by adding r vertices, say 1 2{ , ,..., }rv v v to the graph G and join
them to the vertex with label 1. Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will
receive label 0.
Hence, (1) (0)f fe e = 1 0( ) ( ) 0n E n E r .
Case 2: When 0 1( ) ( ) 1n E n E (say r ).](https://image.slidesharecdn.com/n011138085-151123095857-lva1-app6891/85/On-Embedding-and-NP-Complete-Problems-of-Equitable-Labelings-2-320.jpg)
![On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 82 |Page
The new graph H can be obtained by adding a path of r vertices, say 1 2{ , ,..., }rv v v to the graph G
and join the vertex 1v to the vertex with label p . Assign label p i to the vertex iv for 1,2,...,i r .
Consequently r edges will receive label 1.
Hence, (1) (0)f fe e = 1 0( ) ( ) 0n E r n E .
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for difference
cordial graph. That is, any graph G can be embedded as an induced subgraph of a difference cordial graph.
Definition 2.5: A graph G is called simply sequential if there is a bijection : ( ) ( ) {1,2,..., }f V G E G p q
such that for each edge { } ( )e uv E G one has ( ) ( ) ( )f e f u f v .
In 1983, Bange et al. [7] have introduced the concept of sequential labeling.
Definition 2.6: If a mapping : ( ) ( ) {0,1}f V G E G is such that for each edge e uv , ( ) ( ) ( )f e f u f v
and the condition (1) (1) (0) (0) 1f f f fv e v e holds then G is called total sequential cordial graph.
In 2002, Cahit [8] has introduced the concept of total sequential cordial labeling as a weaker version of
sequential labeling.
Theorem 2.7: Any graph G can be embedded as an induced subgraph of a total sequential cordial graph.
Proof. Label the vertices of graph G in such a way that vertices with label 0 and vertices with label 1 differ by
at most 1.
Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and
in E be the cardinality of set iV and iE respectively.
Case 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 0 to these new vertices. Consequently
2
r
edges will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Case 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 1 to these new vertices. Consequently
2
r
edges will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total sequential
cordial graph. That is, any graph G can be embedded as an induced subgraph of a total sequential cordial graph.
Definition 2.8: Let : ( ) {1,2,..., }f V G p be a bijection. For each edge uv , assign the label 1 if either
( ) | ( )f u f v or ( ) | ( )f v f u and the label 0 otherwise. f is called a divisor cordial labeling if (1) (0) 1f fe e .
A graph with a divisor cordial labeling is called a divisor cordial graph.
In 2011, Varatharajan et al. [9] have introduced the concept of divisor cordial labeling.
Theorem 2.9: Any graph G can be embedded as an induced subgraph of a divisor cordial graph.
Proof. Let G be the given graph with p vertices and q edges. As per definition of divisor cordial labeling
assign labels {1,2,..., }p to the vertices of G .
Let iE be the set of edges with label i while in E be the cardinality of set iE .
Case 1: When 0 1 1n E n E (say r ).](https://image.slidesharecdn.com/n011138085-151123095857-lva1-app6891/85/On-Embedding-and-NP-Complete-Problems-of-Equitable-Labelings-3-320.jpg)
![On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 83 |Page
The new graph H can be obtained by adding r vertices to the graph G , say 1 2, ,..., rv v v and join them
to vertex with label 1. Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will receive
label 1.
Hence, (1) (0)f fe e = 0 1 0n E n E r .
Case 2: When 1 0 1n E n E (say r ).
The new graph H can be obtained by adding a path of r vertices, say 1 2, ,..., rv v v to the graph G and
join the vertex 1v to the vertex with label p . Assign label p i to the vertex iv for 1,2,...,i r .
Consequently r edges will receive label 0.
Hence, (1) (0)f fe e = 0 1 0n E r n E .
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for divisor cordial
graph. That is, any graph G can be embedded as an induced subgraph of a divisor cordial graph.
Definition 2.10: A graph G is said to have a magic labeling with constant C if there exists a bijection
: ( ) ( ) {1,2,..., }f V G E G p q such that ( ) ( ) ( )f u f v f uv C for all ( )uv E G .
Kotzig and Rosa [10] have introduced the concept of magic labeling in 1970 and investigated magic
labeling for some standard graphs.
Definition 2.11: A graph G is said to have a totally magic cordial labeling with constant C if there exists a
mapping : ( ) ( ) {0,1}f V G E G such that ( ) ( ) ( )f u f v f uv C (mod 2) for all ( )uv E G provided the
condition (1) (1) (0) (0) 1f f f fv e v e holds.
In 2002, Cahit [8] has introduced the concept of total magic cordial labeling as a weaker version of
magic labeling.
Theorem 2.12: Any graph G can be embedded as an induced subgraph of a total magic cordial graph.
Proof. Label the vertices of graph G in such a way that (1) (0) 1f fv v and as per definition of total magic
cordial labeling assign edge labels such that ( ) ( ) ( )f u f v f uv C (mod 2).
Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and
in E be the cardinality of set iV and iE respectively.
Case 1: If 0C .
Subcase 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 0 to these new vertices. Consequently
2
r
edges will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Subcase 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 1 to these new vertices. Consequently
2
r
edges will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Case 2: If 1C .
Subcase 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).](https://image.slidesharecdn.com/n011138085-151123095857-lva1-app6891/85/On-Embedding-and-NP-Complete-Problems-of-Equitable-Labelings-4-320.jpg)
![On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 84 |Page
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 1V . Assign label 0 to these new vertices. Consequently
2
r
edges will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Subcase 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 1V . Assign label 1 to these new vertices. Consequently
2
r
edges will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total magic
cordial graph. That is, any graph G can be embedded as an induced subgraph of a total magic cordial graph.
III. NP-Complete Problems
Theorem 3.1: Any planar graph G can be embedded as an induced subgraph of a planar total edge product
cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph.
Proof. If G is planar graph. Then the supergraph H constructed in Theorem 2.2 (corresponding theorems 2.4,
2.7, 2.9 and 2.12) is a planar graph. Hence the result.
Theorem 3.2: Any connected graph G can be embedded as an induced subgraph of a connected total edge
product cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph.
Proof. If G is connected graph. Then the supergraph H constructed in Theorem 2.2 (corresponding theorems
2.4, 2.7, 2.9 and 2.12) is a connected graph. Hence the result.
Theorem 3.3: The problem of deciding whether the chromatic number ( )G k , where 3k , is NP-complete
even for total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic
cordial) graph.
Proof. Let G be a graph with chromatic number ( ) 3G . Let supergraph H constructed in Theorem 2.2 is
total edge product cordial (corresponding theorems 2.4, 2.7, 2.9 and 2.12), which contains G as an induced
subgraph. Then obviously we have ( ) ( )H H . Since the problem of deciding whether the chromatic
number ( )G k , where 3k , is NP-complete by [11]. It follows that deciding whether the chromatic
number ( )G k , where 3k , is NP-complete even for prime cordial graphs.
Theorem 3.4: The problem of deciding whether the clique number ( )G k is NP-complete even when
restricted to total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic
cordial) graph.
Proof. Since the problem of deciding whether the clique number of a graph ( )G k is NP-complete by [11]
and ( ) ( )H G for the supergraph H constructed in Theorem 2.2 is total edge product cordial
(corresponding theorems 2.4, 2.7, 2.9 and 2.12). Hence the result.
Theorem 3.5: The problem of deciding whether the domination number (total domination number) is less than
or equal to k is NP-complete even when restricted to total edge product cordial (difference cordial, total
sequential cordial, divisor cordial, total magic cordial) graph.
Proof. Since the problem of deciding whether the domination number (total domination number) of a graph G
is less than or equal to k is NP-complete as reported in [11] and the supergraph H constructed in Theorem 2.2
is total edge product cordial (corresponding theorems 2.4, 2.7, 2.9 and 2.12) has domination number greater than
or equal to domination number of G . Hence the result.](https://image.slidesharecdn.com/n011138085-151123095857-lva1-app6891/85/On-Embedding-and-NP-Complete-Problems-of-Equitable-Labelings-5-320.jpg)
![On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 85 |Page
IV. Concluding Remarks
Acharya et al. [12] have proved that any graph G can be embedded as an induced subgraph of a
graceful graph. Thus showing the impossibility of obtaining any forbidden subgraph characterization for
graceful graphs on the same line Acharya et al. [13, 14], Anandavally et al. [15], Germina and Ajitha [16],
Vaidya and Vihol [17] and Vaidya and Barasara [18] have discussed embedding and NP-Complete problems in
the context of various graph labeling schemes. However here we have discussed embedding and NP-Complete
problems for total edge product cordial labeling, difference cordial labeling, total sequential cordial labeling,
divisor cordial labeling and total magic cordial labeling.
Acknowledgement
The authors are highly thankful to the anonymous referee for their kind suggestions and comments.
References
[1] J. A. Gallian, A Dynamic Survey of Graph Labeling, The electronic journal of combinatorics, 17, 2014, #DS6.
[2] I. Cahit, Cordial Graphs: A weaker version of graceful and harmonious graphs, Ars Combinatoria, 23, 1987, 201 - 207.
[3] S. K. Vaidya and C. M. Barasara, Edge Product Cordial Labeling of Graphs, Journal of Mathematical and Computational Science,
2(5), 2012, 1436 - 1450.
[4] G. Chartrand and L. Lesniak, Graphs and Digraphs 4/e (Chapman & Hall/CRC Press, 2005).
[5] S. K. Vaidya and C. M. Barasara, Total edge product cordial labeling of graphs, Malaya Journal of Matematik, 3(1), 2013, 55 - 63.
[6] R. Ponraj, S. Sathish Narayanan and R. Kala, Difference Cordial Labeling of Graphs, Global Journal of Mathematical Sciences:
Theory and Practical, 5(3), 2013, 185 - 196.
[7] D. W. Bange, A. E. Barkauskas and P. J. Slater, Simply sequential and graceful graphs, Proceeding of Tenth Southeastern
Conference on Combinatorics, Graph Theory and Computing, Utilitas Mathematica Publishing Inc., Winnipeg, Canada, 1979.
[8] I. Cahit, Some Totally Modular Cordial Graphs, Discussiones Mathematicae Graph Theory, 22, 2002, 247 - 258.
[9] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Divisor cordial graphs, International Journal of Mathematical
Combinatorics, 4, 2011, 15 - 25.
[10] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canadian Mathematical Bulletin, 13(4), 1970, 451 - 461.
[11] M. R. Garey and D. S. Johnson. Computers and Intractability- A guide to the theory of NP-Completeness (W. H. Freeman and
Company, 1979).
[12] B. D. Acharya, S. B. Rao and S. Arumugam, Embeddings and NP-Complete problems for Graceful Graphs, Labeling of Discrete
Structures and Applications (B. D. Acharya, S. Arumugam and A. Rosa (eds.)), Narosa Publishing House, New Delhi, 2008, 57 -
62.
[13] B. D. Acharya, K. A. Germina, K. L. Princy and S. B. Rao, Graph Labelings, Embeddings and NP-Completeness theorems, Journal
of Combinatorial Mathematics and Combinatorial Computation, 67, 2008, 163 - 180.
[14] B. D. Acharya, K. A. Germina, K. L. Princy and S. B. Rao, Embeddings and NP-Completeness theorems for set-graceful,
topologically set-graceful and set-sequential graphs, Discrete Mathematical Sciences and Cryptography, 12(4), 2009, 481 - 487.
[15] T. M. K. Anandavally, S. Arumugam, K. A. Germina and S. B. Rao, On Sum Composite Graphs and Embedding Problems, Journal
of Combinatorial Mathematics and Combinatorial Computing, 69, 2009, 23 - 30.
[16] K. A. Germina and V. Ajitha, Embedding Theorems and NP-Completeness, Proceedings of the National Conference on Operator
Theory and Applications, 2009, 97 - 103.
[17] S. K. Vaidya and P. L. Vihol, Embedding and NP-Complete Problems for 3-Equitable Graphs, International Journal of
Contemporary Advanced Mathematics, 2(1), 2011, 1 - 7.
[18] S. K. Vaidya and C. M. Barasara, Embedding and NP-complete Problems for Some Equitable Labelings, Journal of Applied
Computer Science and Mathematics, 18 (8), 2014, 38 – 42.](https://image.slidesharecdn.com/n011138085-151123095857-lva1-app6891/85/On-Embedding-and-NP-Complete-Problems-of-Equitable-Labelings-6-320.jpg)
On Embedding and NP-Complete Problems of Equitable Labelings
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 80-85 www.iosrjournals.org DOI: 10.9790/5728-11138085 www.iosrjournals.org 80 |Page On Embedding and NP-Complete Problems of Equitable Labelings S. K. Vaidya1 , C. M. Barasara2 1 (Department of Mathematics, Saurashtra University, Rajkot - 360 005, Gujarat, INDIA) 2 (Atmiya Institute of Technology and Science, Rajkot - 360 005, Gujarat, INDIA) Abstract: The embedding and NP-complete problems in the context of various graph labeling schemes were remained in the focus of many researchers. Here we have explored embedding and NP-Complete problems for some variants of cordial labeling. Keywords: Graph labeling, equitable labeling, embedding of graphs, NP-complete problems. Mathematics Subject Classification (2010): 05C78, 05C60. I. Introduction A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain condition or conditions. Labeling of discrete structures is a potential area of research and a comprehensive survey on graph labeling can be found in Gallian [1]. Definition 1.1: For a graph G , a vertex labeling function }1,0{)(: GVf induces an edge labeling function }1,0{)(:* GEf defined as )()()(* vfufuvef . Then f is called cordial labeling of graph G if absolute difference of number of vertices with label 1 and label 0 is at most 1 and absolute difference of number of edges with label 1 and label 0 is at most 1. A graph G is called cordial if it admits a cordial labeling. The concept of cordial labeling was introduced by Cahit [2] in 1987 and in the same paper he presented several results on this newly introduced concept. After this some labelings like prime cordial labeling, A - cordial labeling, product cordial labeling, edge product cordial labeling etc. were also introduced with minor variations in cordial theme. Such labelings are classified as equitable labelings. Definition 1.2: For a graph G , an edge labeling function }1,0{)(: GEf induces a vertex labeling function }1,0{)(:* GVf defined as )}(/)({)(* GEuvuvfuf . Then f is called an edge product cordial labeling of graph G if the absolute difference of number of vertices with label 1 and label 0 is at most 1 and the absolute difference of number of edges with label 1 and label 0 is also at most 1. A graph G is called edge product cordial graph if it admits an edge product cordial labeling. In 2012, the concept of edge product cordial labeling was introduced by Vaidya and Barasara [3]. In the same paper they have investigated edge product cordial labeling for some standard graphs. For any graph G , we introduce following notations: 1. )1(fv = the number of vertices having label 1; 2. )0(fv = the number of vertices having label 0; 3. (1)fe = the number of edges having label 1; 4. (0)fe = the number of edges having label 0; Throughout this discussion we consider simple, finite and undirected graph ))(),(( GEGVG with pGV )( and qGE )( . For all other standard terminology and notation we refer to Chartrand and Lesniak[4].
- 2. On Embedding and NP-Complete Problems of Equitable Labelings DOI: 10.9790/5728-11138085 www.iosrjournals.org 81 |Page II. Embedding of Equitable graphs Definition 2.1: For a graphG , an edge labeling function }1,0{)(: GEf induces a vertex labeling function }1,0{)(:* GVf defined as )}(/)({)(* GEuvuvfuf . Then f is called a total edge product cordial labeling of graph G if (1) (1) (0) (0) 1f f f fv e v e . A graph G is called total edge product cordial graph if it admits a total edge product cordial labeling. The concept of total edge product cordial labeling was introduced by Vaidya and Barasara [5]. Theorem 2.2: Any graph G can be embedded as an induced subgraph of a total edge product cordial graph. Proof. Label the edges of G in such a way that edges with label 1 and edges with label 0 differ by at most 1. Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and in E be the cardinality of set iV and iE respectively. Case 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ). The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 0V . Assign label 0 to these new edges. Consequently 2 r vertices will receive label 0. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1 2 r n V n E n V n E . Case 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ). The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 0V . Assign label 1 to these new edges. Consequently 2 r vertices will receive label 1. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1 2 r n V n E n V n E . Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total edge product cordial graph. That is, any graph G can be embedded as an induced subgraph of a total edge product cordial graph. Definition 2.3: Let : ( ) {1,2,..., }f V G p be a bijection. For each edge uv , assign the label ( ) ( )f u f v . Function f is called a difference cordial labeling if (1) (0) 1f fe e . where (1)fe and (0)fe denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is called a difference cordial graph. The concept of difference cordial labeling was introduced by Ponraj et al. [6] in 2003. Theorem 2.4: Any graph G can be embedded as an induced subgraph of a difference cordial graph. Proof. Let G be the given graph with p vertices and q edges. As per definition of difference cordial labeling assign labels {1,2,..., }p to the vertices of G . Let iE be the set of edges with label i while in E be the cardinality of set iE . Case 1: When 1 0( ) ( ) 1n E n E (say r ). The new graph H can be obtained by adding r vertices, say 1 2{ , ,..., }rv v v to the graph G and join them to the vertex with label 1. Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will receive label 0. Hence, (1) (0)f fe e = 1 0( ) ( ) 0n E n E r . Case 2: When 0 1( ) ( ) 1n E n E (say r ).
- 3. On Embedding and NP-Complete Problems of Equitable Labelings DOI: 10.9790/5728-11138085 www.iosrjournals.org 82 |Page The new graph H can be obtained by adding a path of r vertices, say 1 2{ , ,..., }rv v v to the graph G and join the vertex 1v to the vertex with label p . Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will receive label 1. Hence, (1) (0)f fe e = 1 0( ) ( ) 0n E r n E . Thus in all the possibilities, the constructed supergraph H satisfies the conditions for difference cordial graph. That is, any graph G can be embedded as an induced subgraph of a difference cordial graph. Definition 2.5: A graph G is called simply sequential if there is a bijection : ( ) ( ) {1,2,..., }f V G E G p q such that for each edge { } ( )e uv E G one has ( ) ( ) ( )f e f u f v . In 1983, Bange et al. [7] have introduced the concept of sequential labeling. Definition 2.6: If a mapping : ( ) ( ) {0,1}f V G E G is such that for each edge e uv , ( ) ( ) ( )f e f u f v and the condition (1) (1) (0) (0) 1f f f fv e v e holds then G is called total sequential cordial graph. In 2002, Cahit [8] has introduced the concept of total sequential cordial labeling as a weaker version of sequential labeling. Theorem 2.7: Any graph G can be embedded as an induced subgraph of a total sequential cordial graph. Proof. Label the vertices of graph G in such a way that vertices with label 0 and vertices with label 1 differ by at most 1. Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and in E be the cardinality of set iV and iE respectively. Case 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ). The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 0V . Assign label 0 to these new vertices. Consequently 2 r edges will receive label 0. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1 2 r n V n E n V n E . Case 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ). The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 0V . Assign label 1 to these new vertices. Consequently 2 r edges will receive label 1. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1 2 r n V n E n V n E . Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total sequential cordial graph. That is, any graph G can be embedded as an induced subgraph of a total sequential cordial graph. Definition 2.8: Let : ( ) {1,2,..., }f V G p be a bijection. For each edge uv , assign the label 1 if either ( ) | ( )f u f v or ( ) | ( )f v f u and the label 0 otherwise. f is called a divisor cordial labeling if (1) (0) 1f fe e . A graph with a divisor cordial labeling is called a divisor cordial graph. In 2011, Varatharajan et al. [9] have introduced the concept of divisor cordial labeling. Theorem 2.9: Any graph G can be embedded as an induced subgraph of a divisor cordial graph. Proof. Let G be the given graph with p vertices and q edges. As per definition of divisor cordial labeling assign labels {1,2,..., }p to the vertices of G . Let iE be the set of edges with label i while in E be the cardinality of set iE . Case 1: When 0 1 1n E n E (say r ).
- 4. On Embedding and NP-Complete Problems of Equitable Labelings DOI: 10.9790/5728-11138085 www.iosrjournals.org 83 |Page The new graph H can be obtained by adding r vertices to the graph G , say 1 2, ,..., rv v v and join them to vertex with label 1. Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will receive label 1. Hence, (1) (0)f fe e = 0 1 0n E n E r . Case 2: When 1 0 1n E n E (say r ). The new graph H can be obtained by adding a path of r vertices, say 1 2, ,..., rv v v to the graph G and join the vertex 1v to the vertex with label p . Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will receive label 0. Hence, (1) (0)f fe e = 0 1 0n E r n E . Thus in all the possibilities, the constructed supergraph H satisfies the conditions for divisor cordial graph. That is, any graph G can be embedded as an induced subgraph of a divisor cordial graph. Definition 2.10: A graph G is said to have a magic labeling with constant C if there exists a bijection : ( ) ( ) {1,2,..., }f V G E G p q such that ( ) ( ) ( )f u f v f uv C for all ( )uv E G . Kotzig and Rosa [10] have introduced the concept of magic labeling in 1970 and investigated magic labeling for some standard graphs. Definition 2.11: A graph G is said to have a totally magic cordial labeling with constant C if there exists a mapping : ( ) ( ) {0,1}f V G E G such that ( ) ( ) ( )f u f v f uv C (mod 2) for all ( )uv E G provided the condition (1) (1) (0) (0) 1f f f fv e v e holds. In 2002, Cahit [8] has introduced the concept of total magic cordial labeling as a weaker version of magic labeling. Theorem 2.12: Any graph G can be embedded as an induced subgraph of a total magic cordial graph. Proof. Label the vertices of graph G in such a way that (1) (0) 1f fv v and as per definition of total magic cordial labeling assign edge labels such that ( ) ( ) ( )f u f v f uv C (mod 2). Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and in E be the cardinality of set iV and iE respectively. Case 1: If 0C . Subcase 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ). The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 0V . Assign label 0 to these new vertices. Consequently 2 r edges will receive label 0. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1 2 r n V n E n V n E . Subcase 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ). The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 0V . Assign label 1 to these new vertices. Consequently 2 r edges will receive label 1. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1 2 r n V n E n V n E . Case 2: If 1C . Subcase 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
- 5. On Embedding and NP-Complete Problems of Equitable Labelings DOI: 10.9790/5728-11138085 www.iosrjournals.org 84 |Page The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 1V . Assign label 0 to these new vertices. Consequently 2 r edges will receive label 0. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1 2 r n V n E n V n E . Subcase 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ). The new graph H can be obtained by adding 2 r vertices to the graph G and join them to arbitrary member of 1V . Assign label 1 to these new vertices. Consequently 2 r edges will receive label 1. Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1 2 r n V n E n V n E . Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total magic cordial graph. That is, any graph G can be embedded as an induced subgraph of a total magic cordial graph. III. NP-Complete Problems Theorem 3.1: Any planar graph G can be embedded as an induced subgraph of a planar total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph. Proof. If G is planar graph. Then the supergraph H constructed in Theorem 2.2 (corresponding theorems 2.4, 2.7, 2.9 and 2.12) is a planar graph. Hence the result. Theorem 3.2: Any connected graph G can be embedded as an induced subgraph of a connected total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph. Proof. If G is connected graph. Then the supergraph H constructed in Theorem 2.2 (corresponding theorems 2.4, 2.7, 2.9 and 2.12) is a connected graph. Hence the result. Theorem 3.3: The problem of deciding whether the chromatic number ( )G k , where 3k , is NP-complete even for total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph. Proof. Let G be a graph with chromatic number ( ) 3G . Let supergraph H constructed in Theorem 2.2 is total edge product cordial (corresponding theorems 2.4, 2.7, 2.9 and 2.12), which contains G as an induced subgraph. Then obviously we have ( ) ( )H H . Since the problem of deciding whether the chromatic number ( )G k , where 3k , is NP-complete by [11]. It follows that deciding whether the chromatic number ( )G k , where 3k , is NP-complete even for prime cordial graphs. Theorem 3.4: The problem of deciding whether the clique number ( )G k is NP-complete even when restricted to total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph. Proof. Since the problem of deciding whether the clique number of a graph ( )G k is NP-complete by [11] and ( ) ( )H G for the supergraph H constructed in Theorem 2.2 is total edge product cordial (corresponding theorems 2.4, 2.7, 2.9 and 2.12). Hence the result. Theorem 3.5: The problem of deciding whether the domination number (total domination number) is less than or equal to k is NP-complete even when restricted to total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph. Proof. Since the problem of deciding whether the domination number (total domination number) of a graph G is less than or equal to k is NP-complete as reported in [11] and the supergraph H constructed in Theorem 2.2 is total edge product cordial (corresponding theorems 2.4, 2.7, 2.9 and 2.12) has domination number greater than or equal to domination number of G . Hence the result.
- 6. On Embedding and NP-Complete Problems of Equitable Labelings DOI: 10.9790/5728-11138085 www.iosrjournals.org 85 |Page IV. Concluding Remarks Acharya et al. [12] have proved that any graph G can be embedded as an induced subgraph of a graceful graph. Thus showing the impossibility of obtaining any forbidden subgraph characterization for graceful graphs on the same line Acharya et al. [13, 14], Anandavally et al. [15], Germina and Ajitha [16], Vaidya and Vihol [17] and Vaidya and Barasara [18] have discussed embedding and NP-Complete problems in the context of various graph labeling schemes. However here we have discussed embedding and NP-Complete problems for total edge product cordial labeling, difference cordial labeling, total sequential cordial labeling, divisor cordial labeling and total magic cordial labeling. Acknowledgement The authors are highly thankful to the anonymous referee for their kind suggestions and comments. References [1] J. A. Gallian, A Dynamic Survey of Graph Labeling, The electronic journal of combinatorics, 17, 2014, #DS6. [2] I. Cahit, Cordial Graphs: A weaker version of graceful and harmonious graphs, Ars Combinatoria, 23, 1987, 201 - 207. [3] S. K. Vaidya and C. M. Barasara, Edge Product Cordial Labeling of Graphs, Journal of Mathematical and Computational Science, 2(5), 2012, 1436 - 1450. [4] G. Chartrand and L. Lesniak, Graphs and Digraphs 4/e (Chapman & Hall/CRC Press, 2005). [5] S. K. Vaidya and C. M. Barasara, Total edge product cordial labeling of graphs, Malaya Journal of Matematik, 3(1), 2013, 55 - 63. [6] R. Ponraj, S. Sathish Narayanan and R. Kala, Difference Cordial Labeling of Graphs, Global Journal of Mathematical Sciences: Theory and Practical, 5(3), 2013, 185 - 196. [7] D. W. Bange, A. E. Barkauskas and P. J. Slater, Simply sequential and graceful graphs, Proceeding of Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing, Utilitas Mathematica Publishing Inc., Winnipeg, Canada, 1979. [8] I. Cahit, Some Totally Modular Cordial Graphs, Discussiones Mathematicae Graph Theory, 22, 2002, 247 - 258. [9] R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan, Divisor cordial graphs, International Journal of Mathematical Combinatorics, 4, 2011, 15 - 25. [10] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canadian Mathematical Bulletin, 13(4), 1970, 451 - 461. [11] M. R. Garey and D. S. Johnson. Computers and Intractability- A guide to the theory of NP-Completeness (W. H. Freeman and Company, 1979). [12] B. D. Acharya, S. B. Rao and S. Arumugam, Embeddings and NP-Complete problems for Graceful Graphs, Labeling of Discrete Structures and Applications (B. D. Acharya, S. Arumugam and A. Rosa (eds.)), Narosa Publishing House, New Delhi, 2008, 57 - 62. [13] B. D. Acharya, K. A. Germina, K. L. Princy and S. B. Rao, Graph Labelings, Embeddings and NP-Completeness theorems, Journal of Combinatorial Mathematics and Combinatorial Computation, 67, 2008, 163 - 180. [14] B. D. Acharya, K. A. Germina, K. L. Princy and S. B. Rao, Embeddings and NP-Completeness theorems for set-graceful, topologically set-graceful and set-sequential graphs, Discrete Mathematical Sciences and Cryptography, 12(4), 2009, 481 - 487. [15] T. M. K. Anandavally, S. Arumugam, K. A. Germina and S. B. Rao, On Sum Composite Graphs and Embedding Problems, Journal of Combinatorial Mathematics and Combinatorial Computing, 69, 2009, 23 - 30. [16] K. A. Germina and V. Ajitha, Embedding Theorems and NP-Completeness, Proceedings of the National Conference on Operator Theory and Applications, 2009, 97 - 103. [17] S. K. Vaidya and P. L. Vihol, Embedding and NP-Complete Problems for 3-Equitable Graphs, International Journal of Contemporary Advanced Mathematics, 2(1), 2011, 1 - 7. [18] S. K. Vaidya and C. M. Barasara, Embedding and NP-complete Problems for Some Equitable Labelings, Journal of Applied Computer Science and Mathematics, 18 (8), 2014, 38 – 42.