A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. ...

A connected graph of girth m ≥ 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showe...

We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r + 3, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that have girth at least 2r+3 and no degree one vertices, which is a step towards a recent ...

We show a simple deterministic linear time algorithm for computing the minimum spanning tree of graphs on n vertices with girth at least b(t, lg n), where t ≥ 1 is a constant and b(x, y) is a variant of the inverse of the fast-growing Ackermann function. We also prove: (i) A deterministic linear time algorithm for general graphs follows from an algorithm for graphs with girth at least g for any...

ABSTRACT. A number of recent works have used a variety of combinatorial constructions to derive Tanner graphs for LDPC codes and some of these have been shown to perform well in terms of their probability of error curves and error floors. Such graphs are bipartite and many of these constructions yield biregular graphs where the degree of left vertices is a constant c+1 and that of the right ver...

For a set of integers S, define ( S APk ) to be the k-uniform hypergraph with vertex set S and hyperedges corresponding to the set of all arithmetic progression of length k in S. Similarly, for a graph H, define ( H Kk ) to be the ( k 2 ) -uniform hypergraph on the vertex set E(H) with hyperedges corresponding to the edge sets of all copies of Kk in H. Also, we say that a k-uniform hypergraph h...

Let G be a graph with n vertices, m edges, girth g, and spectral radius μ. Then

Robertson conjectured that the only 3-connected, internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We provide a counterexample to this conjecture.

Let G = (V, E) be a graph. A set M of edges is called a matching in G if each vertex in G belongs to at most one edge from M , and M is a maximal matching if any edgeset M ′, such that M ⊂ M ′, is not a matching in G. If all maximal matchings in G have the same cardinality then G is an equimatchable graph. In this paper we characterize the equimatchable graphs of girth at least five. As a conse...