New Classes of Graceful Unicyclic Graphs

New Classes of Graceful Unicyclic Graphs

Available online at www.sciencedirect.com Electronic Notes in Discrete Mathematics 48 (2015) 27–32 www.elsevier.com/locate/endm New Classes of Grac...

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 48 (2015) 27–32

www.elsevier.com/locate/endm

New Classes of Graceful Unicyclic Graphs Jay Bagga Ball State University, USA.

Laure Pauline Fotso, Pambe Biatch’ Max University of Yaounde I, Cameroon.

S. Arumugam National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India.

Abstract A Cn -unicyclic graph is a unicyclic graph where the cycle has n ≥ 3 vertices. A caterpillar R with spine Pn = v0 v1 . . . vn−1 is denoted by R(v0 v1 . . . vn−1 ). A cycle with a pendant caterpillar is obtained by identifying a vertex of the cycle with a leaf of R(v0 v1 . . . vn−1 ) that is adjacent to v0 (or vn−1 ). In this paper, we investigate the gracefulness of unicyclic graphs with pendant caterpillars at two adjacent vertices of the cycle, and pendant edges at some other vertices of the cycle. Keywords: Unicyclic graph, Canonical labeling, Graceful labeling.

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Introduction

For a a simple graph G = (V, E), a graceful labeling f is a one-to-one map from V to the set {0, 1, · · · , |E(G)|} such that when an edge xy is assigned the label |f (x) − f (y)|, the resulting edge labels are all distinct. A graph G which admits at least one graceful labeling is a graceful graph. The notion of graceful labeling was introduced by Rosa [8], and was motivated by a conjecture of Ringel [8] that states that given any tree T with n edges, the complete graph K2n+1 can be edge-decomposed into 2n + 1 copies of T . Rosa [8] showed that if a tree T with n edges is graceful, then K2n+1 can be so edge-decomposed. Thus Ringel’s conjecture is reduced to proving gracefulness of all trees. Gracefulness of other types of graphs has also been investigated. See Gallian’s survey [6] for details. In particular, we are interested in unicyclic graphs. Rosa [8] proved that the n-cycle Cn is graceful if and only if n ≡ 0 or 3 (mod 4). We are interested in Truszczynski’s conjecture [9], which says that all unicyclic graphs, except cycles Cn with n ≡ 1 or 2 (mod 4) are graceful. Barientos [4] proved that a unicyclic graph in which the deletion of any edge on the cycle results in a caterpillar is graceful. Bagga et al. [1,2,3,7] investigated algorithms to generate all graceful labelings of several classes of unicyclic graphs. In this paper, we present some new classes of graceful unicyclic graphs. We give a brief summary of deﬁnitions and notations used in the paper. For all other standard terminology, we follow Diestel [5]. We consider only connected graphs. A unicyclic graph is a graph with exactly one cycle. Clearly, for unicyclic graphs, the number of vertices is equal to the number of edges. A Cn −unicyclic graph is a unicyclic graph where the cycle has n ≥ 3 vertices. A path on n vertices is denoted Pn (n ≥ 1). A caterpillar is a tree that is either a P2 or has more than two vertices and the removal of its leaves results in a path Pn . This path is called the spine of the caterpillar. A caterpillar R with spine Pn = v0 v1 · · · vn−1 is also denoted R(v0 v1 · · · vn−1 ). A cycle with a pendant caterpillar is obtained by identifying a vertex of the cycle with a leaf of R(v0 v1 · · · vn−1 ) that is adjacent to v0 (or vn−1 ). Rosa[8] proved that every caterpillar is graceful. Since we shall use Rosa’s canonical graceful labeling of a caterpillar, we describe it next. Suppose R is a caterpillar of order p. If p = 2, then label the vertices 1 and 0. For p ≥ 3, let R = R(v0 v1 · · · vn−1 ) (n ≥ 1). Partition the vertices of R in two partite sets A and B where A consists of vertices that are at an even distance from v0 , and B consists of vertices that are at an odd distance from v0 . Clearly vi ∈ A for even i and vi ∈ B for odd i. Label v0 with label p − 1. Label the neighbors of v0 with labels 0, 1, 2 · · · , such that the neighbor v1 gets the largest of these

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labels. Label the unlabeled neighbors of v1 with labels p − 2, p − 3, p − 4, · · · such that the neighbor v2 gets the smallest of these labels. Continue in this fashion until all vertices have labels. This process assigns consecutive labels p − 1, p − 2, p − 3, · · · to vertices in A, and 0, 1, 2, · · · to those in B. It follows that the edges are labeled p − 1, p − 2, · · · , 2, 1 and a graceful labeling is obtained.

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Unicyclic graphs with two pendant caterpillars

In this section, we study gracefulness of unicyclic graphs that have two pendant caterpillars. More precisely, we consider certain unicyclic graphs that have two pendant caterpillars at two adjacent vertices of the cycle. We consider the cases when the length of the cycle is odd and even, separately. 2.1

Graceful C2n+1 −unicyclic graphs

Theorem 2.1 Suppose that G is a C2n+1 −unicyclic graph where the cycle vertices (in order) are x0 , x1 , x2 , · · · , x2n , such that all the vertices of the cycle except two adjacent vertices (say) x0 and x2n have degree two in G, and there are pendant caterpillars R1 and R2 at x0 and x2n such that |E(R2 )| ≥ n − 1. Then G is graceful. Proof. We observe that the graph obtained from G by removing the cycle edge x0 x2n is a caterpillar (say R) of order 2n + 1 + |E(R1 )| + |E(R2 )| = p (say). We perform the canonical labeling of R (described in the previous section), starting at a spine vertex of R1 farthest from x0 , except that we begin with the label p and not p − 1. It follows that the edges are labeled p, p−1, p−2, · · · , 3, 2 in that order. We observe that all the cycle vertices of G are on the spine of the caterpillar R. Furthermore, x0 and x2n are in the same partite set, as are x2 , x4 , · · · , x2n−2 . Suppose that the labels of x0 and x2n are l1 and l2 , respectively. It follows that |l1 − l2 | = n. Since |E(R2 )| ≥ n − 1 and the labels end at 2, n appears as an edge label on an edge of R2 . We now modify the labels of all the vertices starting with the latter vertex (in the order of canonical labeling) on the caterpillar that produced the edge label n. We skip that vertex label and choose the next available one. It follows that the corresponding edge labels get changed to n − 1, n − 2, · · · 2, 1. We now have a graceful labeling of G where the edge x0 x2n on the cycle is labeled n. This completes the proof. 2 The result above can be generalized to the case when there are two pendant

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caterpillars as above, and the remaining cycle vertices have any number of pendant P2 s. We present this next. Theorem 2.2 Suppose that G is a C2n+1 -unicyclic graph where the cycle vertices (in order) are x0 , x1 , x2 , · · · , x2n , such that two adjacent vertices on the cycle (say x0 and x2n ) have pendant caterpillars R1 and R2 , while every other vertex on the cycle has any number of pendant P2 s. Suppose that the total number of pendant P2 s at vertices x1 , x3 , · · · , x2n−1 is t. If |E(R2 )| ≥ n+t−1, then G is graceful. Proof. This proof is similar to that of Theorem 2.1. As before, we do a modiﬁed canonical labeling of the caterpillar R obtained from G by removing the cycle edge x0 x2n to obtain edge labels p, p − 1, p − 2, · · · , 3, 2, where p is the order of G, and the edges of R2 have the lowest labels. In this case there are n + 1 cycle vertices of G that are in the same partite set of R, as are the t leaves of the P2 s pendant at x1 , x3 , · · · , x2n−1 . Hence the absolute diﬀerence between the labels of x0 and x2n is n + t. Since |E(R2 )| ≥ n + t − 1, one of the edge labels of r2 is also n + t. We modify the labeling as above to obtain a graceful labeling of G, with the edge x0 x2n getting the label n + t. This completes the proof. 2 2.2

Graceful C2n −unicyclic graphs

We now discuss the case of a C2n −unicyclic graph, with two caterpillars attached to adjacent vertices of the cycle. Theorem 2.3 Suppose that G is a C2n -unicyclic graph where the cycle vertices (in order) are x0 , x1 , x2 , x3 , · · · , x2n−1 , such that two adjacent vertices on the cycle (say x0 and x2n−1 ) have pendant caterpillars R1 and R2 respectively, while every other vertex on the cycle has any number of pendant P2 s. Suppose that the total number of pendant P2 s at vertices x1 , x3 , · · · , x2n−3 is t. If |E(R2 )| = n + t − 2, then G is graceful. Proof. Let r1 denote |E(R1 )|. Also, let t0 denote the number of pendant P2 s at vertices x2 , x4 , · · · , x2n−2 . The number (say) p of vertices in G is 2n+t+t0 +|E(R1 )|+|E(R2 )| = 2n+t+t0 +r1 +n+t−2 = 3n+r1 +2t+t0 −2 = p. As in the odd case above, we perform a canonical graceful labeling of the caterpillar R (say) obtained from G by removing the cycle edge x0 x2n−1 , where we begin with the label p at the vertex on the spine of R1 that is farthest from

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the cycle. Then the edges of this caterpillar will be labeled p, p − 1, · · · , 2 as follows: • the edges of R1 are labeled p, p − 1, · · · , p − r1 + 1 • the 2n − 1 + t + t0 cycle edges of G and the pendant edges (P2 s) are labeled (in order) p − r1 , p − r1 − 1, · · · , p − r1 − (2n − 2 + t + t0 ), where p − r1 − (2n − 2 + t + t0 ) = 3n + r1 + 2t + t0 − 2 − r1 − (2n − 2 + t + t0 ) = n + t • the edges of R2 are labeled n + t − 1, n + t − 2, · · · , 2. Let x denote the vertex of R2 that is adjacent to x2n−1 . We observe that the vertices x0 , x2 , · · · , x2n−2 , the leaves of pendant P2 s at x1 , x3 , · · · , x2n−3 and x are in the same partite set of R . Hence the absolute diﬀerence in the labels of x0 and x2n−2 is n + t − 1, while that between the labels of x2n−2 and x is 1. We now swap the labels of x2n−1 and x. This changes the label of the edge x2n−1 x to 1 and that of the edge x2n−1 x0 of G to n + t respectively. Now we relabel the vertices of R2 except x2n−1 by performing a canonical labeling by starting at x with the current label of x. This relabels the edges with labels n + t − 1, n + t − 2, · · · , 2. We now get a graceful labeling of G. 2

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Summary

In this paper, we have studied unicyclic graphs that have two caterpillars pendant at two adjacent vertices of the cycle, and edges pendant at other vertices. Under certain conditions on the size of one of the caterpillars, we show that such graphs are graceful. Our future work will focus on gracefulness of unicyclic graphs with caterpillars attached to all the vertices of the cycle. We will also investigate the outcome of attaching to the vertices of the cycle trees that may not be caterpillars. We hope these studies will take us some steps closer to the proof of Truszczynski’s conjecture.

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References [1] Jay Bagga, Adrian Heinz and M. Mahbubul Majumder, An Algorithm for Graceful Labelings of Cycles, Congr. Numer., 186 (2007), 57–63. [2] Jay Bagga, Adrian Heinz and M. Mahbubul Majumder, Properties of Graceful Labelings of Cycles, Congr. Numer., 188 (2007), 109–115. [3] Jay Bagga, Graceful Labelings - Properties and Algorithms, Graph Theory Research Directions, Eds. Pratima Panigrahi and S.B. Rao, Narosa Publishing House, New Delhi, (2011), 49–58. [4] C. Barrientos, Graceful graphs with pendant edges, Australas. J. Combin., 33 (2005), 99–107. [5] R. Diestel, Graph Theory, Electronic Edition 2005, Springer-Verlag Heidelberg, New York (2005). [6] J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal Combinatorics, 17 (2014), #DS6 [7] Pambe Biatch’ Max, Jay Bagga and Laure Pauline Fotso, An Algorithm for Graceful Labelings of Certain Unicyclic Graphs, VNU Journal of Science: Comp. Science & Com. Eng., 30(3) (2014), 1–11. [8] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967), 349–355. [9] M. Truszczynski, Graceful unicyclic graphs, Demonstatio Mathematica, 17 (1984), 377–387.