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Super total graceful graphs
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Electronic Notes in Discrete Mathematics 48 (2015) 301–304 www.elsevier.com/locate/endm
Super total graceful graphs S. P. Subbiah Department of Mathematics Mannar Thirumalai Naicker College Madurai- 625 004, Tamil Nadu, India.
J. Pandimadevi Department of Mathematics EMG Yadava Women’s College Madurai- 625 014, Tamilnadu, India.
R. Chithra Department of Mathematics The Madura College, Madurai-625 011 Tamil Nadu, India.
Abstract A total graceful labeling (T GL) of a (p, q) graph G is a bijection f from V (G) ∪ E(G) to the set {1, 2, 3, . . . , p + q} so that f (uv) = |f (u) − f (v)| for all uv ∈ E(G). A T GL is called a super if f (E) = {1, 2, 3, . . . , q}. A graph that admits a T GL is called a total graceful graph (T GG) and a graph that admits a super total graceful labeling is called a super total graceful graph (ST GG). In this paper, we show that some trees are super total graceful. Keywords: graceful labeling, total graceful labeling, super total graceful labeling, super total graceful graph.
http://dx.doi.org/10.1016/j.endm.2015.05.045 1571-0653/© 2015 Elsevier B.V. All rights reserved.
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S.P. Subbiah et al. / Electronic Notes in Discrete Mathematics 48 (2015) 301–304
Let G = (V, E) be a graph with vertex set V (G) and edge set E(G). A graph G is a (p, q) graph if |V (G)| = p and |E(G)| = q. A function f is called a graceful labeling of G if f : V (G) → {0, 1, 2, 3, . . . , q} is injective and the induced function f ∗ : E(G) → {1, 2, 3, . . . , q} defined by f ∗ (uv) = |f (u)−f (v)| is bijective. This type of graph labeling was first introduced by Rosa [8] in 1967 as β-valuation and was used as a tool for decomposing a complete graph into isomorphic subgraphs. Even though the concept of a β-valuation was introduced by Rosa, in 1972, Golomb [3] called such labeling as a “graceful labeling”. The concept of a graceful labeling was introduced to attack Ringel’s famous conjecture that K2n+1 could be decomposed into 2n+1 subgraphs that are all isomorphic to a given tree with n edges. Graceful labeling has been applied to areas such as coding theory, radar, radio astronomy, and circuit design. Rosa proved that if G is graceful and all of its vertices are of even degrees, then |E(G)| ≡ 0 or 3(mod 4). Cycle Cn are graceful if and only if n ≡ 0 or 3(mod 4); Paths Pn ; wheels Wn and complete bipartite graphs Km,n are graceful whereas complete graphs Kn are graceful if and only if n ≤ 4 [3]. It has been conjectured that all trees are graceful. It has been shown that trees with at most 35 vertices are graceful. Although this conjecture has been the focus of more than 800 papers, it is still an open problem. More than 1700 papers have been published on the subject of graph labeling. Many of the results on graph labeling have been collected and updated in a survey by Gallian [2]. One can refer to this survey for more information about the subject. Many variations of graceful labeling have been introduced by researchers, for example see [1,2,4,5,6]. In this article, we study a variation of graceful labeling called super total graceful labeling of graphs introduced in [9]. A total graceful labeling (T GL) of a (p, q) graph G is a bijection f : V (G) ∪ E(G) → {1, 2, 3, . . . , p + q} such that f (uv) = |f (u) − f (v)| for all uv ∈ E(G). A T GL is called super if f (E) = {1, 2, 3, . . . , q}. A graph that admits a T GL is called a total graceful graph (T GG) and a graph that admits a super total graceful labeling is called a super total graceful graph (ST GG). Let P1 and P2 be paths on n vertices with vertex sets {u1 , u2 , . . . , un } and {v1 , v2 , . . . , vn } respectively. If n is even, adjoin u n2 +1 and v n2 ; if n is odd, adjoin u n+1 and v n+1 . The resulting graph is called an H-graph. 2
1 2 3
2
Email: [email protected] Email: pandimadevi87@yahoo. Email: [email protected]
S.P. Subbiah et al. / Electronic Notes in Discrete Mathematics 48 (2015) 301–304
303
Theorem 1 The graph H ◦K1 is a super total graceful graph for any H-graph H. The subdivision graph S(G) of a graph G is the graph obtained by replacing every edge of G with two edges, which is achieved by introducing a new vertex w such that uw, wv ∈ E(S(G)) for each uv ∈ E(G). Theorem 2 The graph S(K1,n ) is a super total graceful graph for any n ≥ 1. Theorem 3 The graph S(Pn ◦K1 ) is a super total graceful graph for all n ≥ 1. A graph obtained by duplicating an edge e = uv by an edge e = u v from a graph G is a new graph H such that V (H) = V (G) ∪ {u , v } and E(H) = E(G) ∪ {u w : w ∈ N (u) − {v}} ∪ {v w : w ∈ N (v) − {u}} ∪ u v . We discuss the super total gracefulness of graphs obtained by duplicating an edge of a path. Theorem 4 If G is a super total graceful graph, then q ≤ p − 1. Proof. Let f be a super total graceful labeling of the graph G. Then f (V ) = {q + 1, q + 2, · · · , q + p}. Thus the label of any edge is at most p − 1 and so q ≤ p − 1. 2 In [7], it was conjectured that every tree is super total graceful. We state without proof the following theorem. Theorem 5 Let G be a graph obtained by duplicating an edge e of a path Pn , n ≥ 3. Then G is super total graceful if and only if e is a pendant edge of Pn .
References [1] R. Frucht, Graceful numbering of wheels and related graphs, Ann. N. Y. Acad. Sci., 319 (1979), 219–229. [2] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin., 16 (2013), #DS6. [3] S. W. Golomb, How to number a graph, Graph Theory and Computing, Academic Press, New York, (1972), 23–37. [4] R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Disc. Meth., 1(4) (1980), 382-404. [5] A. Graf, A new graceful labeling for pendant graphs, Aequat. Math., 87 (2014), 135–145.
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S.P. Subbiah et al. / Electronic Notes in Discrete Mathematics 48 (2015) 301–304
[6] S. P. R. Hebbare, Graceful cycles, Util. Math., 7 (1976), 307–317. [7] J. Hopscroft and M. S. Krishnamoorthy, On harmonious coloring of graphs, SIAM J. Alg. Discrete Math., 4 (1983), 306–311. [8] A. Rosa, On certain valuations of the vertices of a graph, Theory of graphs (Internat. Symposium, Rome, July 1996), Gordan and Breach, New York, Dunod, Paris (1967), 349–355. [9] S. P. Subbiah and J. Pandimadevi, Super total graceful graphs and a tree conjecture, Communicated.
Electronic Notes in Discrete Mathematics 48 (2015) 301–304 www.elsevier.com/locate/endm
Super total graceful graphs S. P. Subbiah Department of Mathematics Mannar Thirumalai Naicker College Madurai- 625 004, Tamil Nadu, India.
J. Pandimadevi Department of Mathematics EMG Yadava Women’s College Madurai- 625 014, Tamilnadu, India.
R. Chithra Department of Mathematics The Madura College, Madurai-625 011 Tamil Nadu, India.
Abstract A total graceful labeling (T GL) of a (p, q) graph G is a bijection f from V (G) ∪ E(G) to the set {1, 2, 3, . . . , p + q} so that f (uv) = |f (u) − f (v)| for all uv ∈ E(G). A T GL is called a super if f (E) = {1, 2, 3, . . . , q}. A graph that admits a T GL is called a total graceful graph (T GG) and a graph that admits a super total graceful labeling is called a super total graceful graph (ST GG). In this paper, we show that some trees are super total graceful. Keywords: graceful labeling, total graceful labeling, super total graceful labeling, super total graceful graph.
http://dx.doi.org/10.1016/j.endm.2015.05.045 1571-0653/© 2015 Elsevier B.V. All rights reserved.
302
S.P. Subbiah et al. / Electronic Notes in Discrete Mathematics 48 (2015) 301–304
Let G = (V, E) be a graph with vertex set V (G) and edge set E(G). A graph G is a (p, q) graph if |V (G)| = p and |E(G)| = q. A function f is called a graceful labeling of G if f : V (G) → {0, 1, 2, 3, . . . , q} is injective and the induced function f ∗ : E(G) → {1, 2, 3, . . . , q} defined by f ∗ (uv) = |f (u)−f (v)| is bijective. This type of graph labeling was first introduced by Rosa [8] in 1967 as β-valuation and was used as a tool for decomposing a complete graph into isomorphic subgraphs. Even though the concept of a β-valuation was introduced by Rosa, in 1972, Golomb [3] called such labeling as a “graceful labeling”. The concept of a graceful labeling was introduced to attack Ringel’s famous conjecture that K2n+1 could be decomposed into 2n+1 subgraphs that are all isomorphic to a given tree with n edges. Graceful labeling has been applied to areas such as coding theory, radar, radio astronomy, and circuit design. Rosa proved that if G is graceful and all of its vertices are of even degrees, then |E(G)| ≡ 0 or 3(mod 4). Cycle Cn are graceful if and only if n ≡ 0 or 3(mod 4); Paths Pn ; wheels Wn and complete bipartite graphs Km,n are graceful whereas complete graphs Kn are graceful if and only if n ≤ 4 [3]. It has been conjectured that all trees are graceful. It has been shown that trees with at most 35 vertices are graceful. Although this conjecture has been the focus of more than 800 papers, it is still an open problem. More than 1700 papers have been published on the subject of graph labeling. Many of the results on graph labeling have been collected and updated in a survey by Gallian [2]. One can refer to this survey for more information about the subject. Many variations of graceful labeling have been introduced by researchers, for example see [1,2,4,5,6]. In this article, we study a variation of graceful labeling called super total graceful labeling of graphs introduced in [9]. A total graceful labeling (T GL) of a (p, q) graph G is a bijection f : V (G) ∪ E(G) → {1, 2, 3, . . . , p + q} such that f (uv) = |f (u) − f (v)| for all uv ∈ E(G). A T GL is called super if f (E) = {1, 2, 3, . . . , q}. A graph that admits a T GL is called a total graceful graph (T GG) and a graph that admits a super total graceful labeling is called a super total graceful graph (ST GG). Let P1 and P2 be paths on n vertices with vertex sets {u1 , u2 , . . . , un } and {v1 , v2 , . . . , vn } respectively. If n is even, adjoin u n2 +1 and v n2 ; if n is odd, adjoin u n+1 and v n+1 . The resulting graph is called an H-graph. 2
1 2 3
2
Email: [email protected] Email: pandimadevi87@yahoo. Email: [email protected]
S.P. Subbiah et al. / Electronic Notes in Discrete Mathematics 48 (2015) 301–304
303
Theorem 1 The graph H ◦K1 is a super total graceful graph for any H-graph H. The subdivision graph S(G) of a graph G is the graph obtained by replacing every edge of G with two edges, which is achieved by introducing a new vertex w such that uw, wv ∈ E(S(G)) for each uv ∈ E(G). Theorem 2 The graph S(K1,n ) is a super total graceful graph for any n ≥ 1. Theorem 3 The graph S(Pn ◦K1 ) is a super total graceful graph for all n ≥ 1. A graph obtained by duplicating an edge e = uv by an edge e = u v from a graph G is a new graph H such that V (H) = V (G) ∪ {u , v } and E(H) = E(G) ∪ {u w : w ∈ N (u) − {v}} ∪ {v w : w ∈ N (v) − {u}} ∪ u v . We discuss the super total gracefulness of graphs obtained by duplicating an edge of a path. Theorem 4 If G is a super total graceful graph, then q ≤ p − 1. Proof. Let f be a super total graceful labeling of the graph G. Then f (V ) = {q + 1, q + 2, · · · , q + p}. Thus the label of any edge is at most p − 1 and so q ≤ p − 1. 2 In [7], it was conjectured that every tree is super total graceful. We state without proof the following theorem. Theorem 5 Let G be a graph obtained by duplicating an edge e of a path Pn , n ≥ 3. Then G is super total graceful if and only if e is a pendant edge of Pn .
References [1] R. Frucht, Graceful numbering of wheels and related graphs, Ann. N. Y. Acad. Sci., 319 (1979), 219–229. [2] J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin., 16 (2013), #DS6. [3] S. W. Golomb, How to number a graph, Graph Theory and Computing, Academic Press, New York, (1972), 23–37. [4] R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Disc. Meth., 1(4) (1980), 382-404. [5] A. Graf, A new graceful labeling for pendant graphs, Aequat. Math., 87 (2014), 135–145.
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S.P. Subbiah et al. / Electronic Notes in Discrete Mathematics 48 (2015) 301–304
[6] S. P. R. Hebbare, Graceful cycles, Util. Math., 7 (1976), 307–317. [7] J. Hopscroft and M. S. Krishnamoorthy, On harmonious coloring of graphs, SIAM J. Alg. Discrete Math., 4 (1983), 306–311. [8] A. Rosa, On certain valuations of the vertices of a graph, Theory of graphs (Internat. Symposium, Rome, July 1996), Gordan and Breach, New York, Dunod, Paris (1967), 349–355. [9] S. P. Subbiah and J. Pandimadevi, Super total graceful graphs and a tree conjecture, Communicated.