Solving Graph Coloring Problems with Abstraction and Symmetry

This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number $R(4,3,3)=30$. The number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found "soon". Yet, its precise value has remained unknown for more than 50 years. We illustrate our approach by showing that: (1) there are precisely 78{,}892 $(3,3,3;13)$ Ramsey colorings; and (2) if there exists a $(4,3,3;30)$ Ramsey coloring then it is (13,8,8) regular. Specifically each node has 13 edges in the first color, 8 in the second, and 8 in the third. We conjecture that these two results will help provide a proof that no $(4,3,3;30)$ Ramsey coloring exists implying that $R(4,3,3)=30$

Distance Magic Graphs - a Survey

Let <i>G = (V;E)</i> be a graph of order n. A bijection <i>f : V → {1, 2,...,n} </i>is called <i>a distance magic labeling </i>of G if there exists a positive integer k such that <i>Σ f(u) = k </i> for all <i>v ε V</i>, where <i>N(v)</i> is the open neighborhood of v. The constant k is called the magic constant of the labeling f. Any graph which admits <i>a distance magic labeling </i>is called a distance magic graph. In this paper we present a survey of existing results on distance magic graphs along with our recent results,open problems and conjectures.DOI : http://dx.doi.org/10.22342/jims.0.0.15.11-2

Connectivity and other invariants of generalized products of graphs

Figueroa-Centeno et al. [4] introduced the following product of digraphs let D be a digraph and let G be a family of digraphs such that V (F) = V for every F¿G. Consider any function h:E(D)¿G. Then the product D¿hG is the digraph with vertex set V(D)×V and ((a,x),(b,y))¿E(D¿hG) if and only if (a,b)¿E(D) and (x,y)¿E(h(a,b)). In this paper, we deal with the undirected version of the ¿h-product, which is a generalization of the classical direct product of graphs and, motivated by the ¿h-product, we also recover a generalization of the classical lexicographic product of graphs, namely the °h-product, that was introduced by Sabidussi in 1961. We provide two characterizations for the connectivity of G¿hG that generalize the existing one for the direct product. For G°hG, we provide exact formulas for the connectivity and the edge-connectivity, under the assumption that V (F) = V , for all F¿G. We also introduce some miscellaneous results about other invariants in terms of the factors of both, the ¿h-product and the °h-product. Some of them are easily obtained from the corresponding product of two graphs, but many others generalize the existing ones for the direct and the lexicographic product, respectively. We end up the paper by presenting some structural properties. An interesting result in this direction is a characterization for the existence of a nontrivial decomposition of a given graph G in terms of ¿h-product.Postprint (author's final draft