The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede’s relatively di...

In this article current directions in solving Lovász’s problem about the existence of Hamilton cycles and paths in connected vertex-transitive graphs are given. © 2009 Elsevier B.V. All rights reserved. 1. Historical motivation In 1969, Lovász [59] asked whether every finite connected vertex-transitive graph has a Hamilton path, that is, a simple path going through all vertices, thus tying toge...

All graphs considered in this paper are finite and undirected, without loops or multiple edges. Let G = (V, E) be a graph. Throughout this paper, let m and n denote the numbers of edges and vertices of graph G, respectively. A connected graph is distance-hereditary if the distance between every two vertices in a connected induced subgraph is the same as that in the original graph. Distance-here...

In this paper, a new idea to configure special graph from the discrete topological space is given. Several properties and bounds of are introduced. Such that if order non-empty set equals two, then isomorphic complete graph. If three, complement cycle Our has induced subgraphs with or more. It also subgraph. addition, clique number obtained. The proved simple, undirected, connected no pendant v...

A dominating set D ⊆ V (G) of a graph G is a set such that each vertex v ∈ V (G) is either in the set or adjacent to a vertex in the set. Matheson and Tarjan (1996) proved that any n-vertex plane triangulation has a dominating set of size at most n/3, and conjectured a bound of n/4 for n sufficiently large. King and Pelsmajer recently proved this for graphs with maximum degree at most 6. Plumme...

A classic theorem of Erdős and Pósa states that there exists a constant c such that for all positive integers k and graphs G, either G contains k vertex disjoint cycles, or there exists a subset of at most ck log k vertices intersecting every cycle of G. We consider the following generalization of the problem: fix a subset S of vertices of G. An S-cycle is a cycle containing at least one vertex...

This paper was designed to provide an introduction to the vector spaces used in graph theory. In graph theory, an area of mathematics, a cycle space is a vector space defined from an undirected graph; elements of the cycle space represent formal combinations of cycles in the graph. Cycle spaces allow one to use the tools of linear algebra to study graphs. A cycle basis is a set of cycles that g...

An N -angulation is a finite simple plane graph such that each face is bounded by a cycle of length N , where N ≥ 3 is an integer. We consider diagonal transformations in N -angulations which consist of ⌊ 2 ⌋ kinds of operations. In the literature, Wagner proved that any two 3angulations (which are often called triangulations) with the same number of vertices can be transformed into each other ...