The girth is the length of shortest cycle in a graph. Recently, the problem for computing the girth of a graph have been studied well. Itai and Rodeh developed the first non trivial algorithm for computing girth in general graphs. Their algorithm runs in O (nm) time in the worst case and it runs in O(n) time on average case. For unweighted planer graphs, Djidjev gave a O ( n log n ) time algori...

The problem of colouring the square of a graph naturally arises in connection with the distance labelings, which have been studied intensively. We consider this problem for sparse subcubic graphs. We show that the choosability χ (G2) of the square of a subcubic graph G of maximum average degree d is at most four if d < 24/11 and G does not contain a 5-cycle, χ (G2) is at most five if d < 7/3 an...

For a graph G, the Merrifield-Simmons index i(G) is defined as the total number of independent sets of the graph G. Let G(n, l, k) be the class of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. In this paper, we characterize the unique unicyclic graph possessing prescribed girth and pendent vertices with the maximal Merrifield-Simmons index among all graphs ...

Let G = (V,E) be a graph of order p and size q. It is known that if G is super edge-magic graph then q ≤ 2p− 3. Furthermore, if G is super edge-magic and q = 2p− 3, then the girth of G is 3. It is also known that if the girth of G is at least 4 and G is super edge-magic then q ≤ 2p − 5. In this paper we show that there are infinitely many graphs which are super edge-magic, have girth 5, and q =...

He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching. c © 2008 Elsevier B.V. All rights reserved.

We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers.

We give an O(n log n) algorithm for computing the girth (shortest cycle) of an undirected n-vertex planar graph. Our solution extends to any graph of bounded genus. This improves upon the best previously known algorithms for this problem.

W. He et al. showed that a planar graph of girth 11 can be decomposed into a forest and a matching. D. Kleitman et al. proved the same statement for planar graphs of girth 10. We further improve the bound on girth to 9. c © 2007 Elsevier Ltd. All rights reserved.