An edge-coloring of a graph is an assignment of colors to its edges such that no adjacent edges have the same color. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs have been introduced by Földes and Hammer [4] and it is a well-studied class of graphs. However, the problem of deciding the chromatic index of any split graph remains unsolv...

 In this paper, we introduce a suitable generalization of Cayley graphs that is defined over polygroups (GCP-graph) and give some examples and properties. Then, we mention a generalization of NEPS that contains some known graph operations and apply to GCP-graphs. Finally, we prove that the product of GCP-graphs is again a GCP-graph.

To compare several promising product designs, manufacturers must measure their performance under multiple environmental conditions. In many applications, a product design is considered to be seriously flawed if its performance is poor under any level of the environmental factor. For example, if a particular automobile battery design does not function well under some temperature conditions, then...

In [5] Abbott and Katchalski ask if there exists a constant c > 0 such that for every d ≥ 2 there is a snake (cycle without chords) of length at least c3 in the product of d copies of the complete graph K3. We show that the answer to the above question is positive, and that in general for any odd integer n there is a constant cn such that for every d ≥ 2 there is a snake of length at least cnn ...

A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their compliments. Then we split characterizing edge-minimal 2-self-centered graphs into two cases. First, we characterize edge-minimal 2-self-centered graphs without triangles by introducing intersecting bi-i...

Let H be a graph with a designated vertex s, where edges are weighted by nonnegative reals. Splitting edges e = {u, s} and e′ = {s, v} at s is an operation that reduces the weight of each of e and e′ by a real δ > 0 while increasing the weight of edge {u, v} by δ. It is known that all edges incident to s can be split off while preserving the edge-connectivity of H and that such a complete split...

A linear-time algorithm for decomposing a graph into split components is presented. The algorithm uses a new graph transformation technique to gradually transform the given graph so that every split component in it is transformed into a subgraph with very simple structure which can be easily identified. Once the split components are determined, the triconnected components of the graph are easil...

Rank-width is a graph complexity measure that has many structural properties. It is known that the rank-width of an undirected graph is the maximum over all induced prime graphs with respect to split decomposition and an undirected graph has rank-width at most 1 if and only if it is a distance-hereditary graph. We are interested in an extension of these results to directed graphs. We give sever...

A dominating set D ⊆ V is a split dominating set of a graph G = (V,E) if the induced subgraph of 〈V −D〉 is disconnected. The split domination number γs(G) is the minimum cardinality of a split dominating set of a graph G. In this article, we establish some results on split domination number of Pm⊕ Pn, Pm ⊕ Cn and Cm ⊕ Cn.