In this paper we study threshold-coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. A pair of vertices with a small difference in their colors implies that the edge between them is present, while a pair of vertices with a big color difference implies that the edge is absent. Not all planar graphs are thre...

Let G be a series-parallel graph with integer edge weights. A p-coloring of G is a mapping of vertices of G into Zp (ring of integers modulo p) so that the distance between colors of adjacent vertices u and v is at least the weight of the edge uv. We describe a quadratic time p-coloring algorithm where p is either twice the maximum edge weight or the largest possible sum of three weights of edg...

Abstract. A skew edge coloring of a graph G is defined to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, an algorithm is determined for skew edge coloring of circular ladder graphs. Alsothe skew chromatic index of circular la...

Whitney’s Broken-cycle Theorem states the chromatic polynomial of a graph as a sum over special edge subsets. We give a definition of cycles in hypergraphs that preserves the statement of the theorem there.

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs are equienergetic if they have the same energy. We construct infinite families of graphs equienergetic with edge-deleted subgraphs.

This paper compares several published algorithms which list all of the maximal cliques of a graph. A clique is a complete subgraph, i.e., a set of vertices and edges such that every pair of vertices is joined by an edge. A maximal clique is a clique which is not a proper subgraph of a larger clique. A closely related concept is that of a maximal independent set (MIS), where every pair of vertic...

A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is known that a plane graph of minimum face size 5 contains light paths and a light pentagon. In this paper we show that every plane graph of minimum face size 5 contains also a light star K1,3 and we present a structural result concerning the existence of a pair of adjacent ...

A skew edge coloring of a graph G is defined to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, skew edge coloring of certain classes of graphs are determined. Furthermore, the skew chromatic index of those graphs is obtained ...

The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper, we study the behavior of the reformulated first Zagreb index and apply our results to different chemically interesting molecular graphs and nano-...

A spanning subgraph S = (V , E ) of a connected graph G = (V , E) is an (x + c)-spanner if for any pair of vertices u and v, dS(u, v) ≤ dG(u, v) + c where dG and dS are the usual distance functions in G and S, respectively. The parameter c is called the delay of the spanner. We study edge-disjoint spanners in graphs in multi-dimensional tori. We show that each two-dimensional torus has a set of...