In this paper we will investigate the edge-chromatic automorphism groups for two classes of graphs. Chromatic automorphism groups have been studied by a number of people, including Chvatal [4] and Baumann et al. [l, 21. Characterizations will be given for these groups. Note that all arithmetic will be done modp, p not necessarily prime. The particular p will be clear from the context. These gra...

The problems under consideration here were united for the first time by the general term “The Three Color Problem” in Ore’s book [7], where the author was interested in the description of planar 3-colorable graphs. Our graph theoretic terminology is that of Harary [4], except that we use vertices and edges instead of points and lines, respectively. The graphs considered here are planar and have...

If G and H are graphs, define the Ramsey number r(G, H) to be the least number p such that if the edges of the complete graph Kp are colored red and blue (say), either the red graph contains G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G . The following result is proved : Let G and H have k and I points respectively and have point independence nu...

A sum graph is a graph for which there is a labeling of its vertices with positive integers so that two vertices are adjacent if and only if the sum of their labels is the label of another vertex. Integral sum graphs are defined similarly, except that the labels may be any integers. These concepts were first introduced by Harary, who provided examples of such graphs of all orders. The family of...

The Catania Research Group develops scientific researches mainly in Graph and Hypergraph Theory and Design Theory. These theories are of great interest not only for their contribution to Discrete Mathematics but also for their applications in Coding Theory, Statistics and Computer Science. The CRG participated in the international Panorama, not only for the developed researches (see the referen...

Topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. One of the useful indic...

We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs. The distinguishing number of a graph is the minimum number of colors needed to label the vertices so that the only color pre...

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs G and H, the Frucht-Harary corona product G ◦H is obtained by taking |G| copies of the cone K1 + H and by identifying the conical vertices according to G. Our work explores conditions under which the coro...

We consider bipartite graphs on m red points and n blue points, where m ^ n, and prove that, for any fixed k, almost all such graphs (labelled or unlabelled) are k-connected as n -» oo, provided m > C logn, where C depends on k. If Tmn is the number of such unlabelled graphs, we show that Tmn ~ 2 /(m \n!). If T'mn is the number of such unlabelled graphs with the colours removed, then T'mn ~ Tmn...

We study 0-1 laws for extensions of first-order logic by Lindström quantifiers. We state sufficient conditions on a quantifier Q expressing a graph property, for the logic FO[Q] – the extension of first-order logic by means of the quantifier Q – to have a 0-1 law. We use these conditions to show, in particular, that FO[Rig], where Rig is the quantifier expressing rigidity, has a 0-1 law. We als...