The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced path. Here are two examples: the 16-vertex Clebsch graph, and the graph obtained from a complete bipartite graph by subdividing a perfect matching. We show that every connected...

A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one rainbow path. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Between them, Chakraborty et al. [J. Comb. O...

A Hamiltonian cycle is a closed path through all the vertices of a graph. Since discovering whether a graph has a Hamiltonian path or a Hamiltonian cycle are both NP-complete problems, researchers concentrated on formulating sufficient conditions that ensure Hamiltonicity of a graph. A Information Processing Letters 94(2005), 37-51] presents distance based sufficient conditions for the existenc...

For a given graph G and a positive integer r the r-path graph, Pr(G), has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r − 1, and their union forms either a cycle or a path of length k+1 in G. Let P k r (G) be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgra...

We determine the computational complexity of several dmision p r o b lerns related to hamiltonian-connected graphs. In particular, we show that it is NP-complete to determine whether a graph G is harniltonian-connected. We also show that it is NP-complete to determine whether G is hamiltonian-connerted from a distinguished vertex u . Lastly, we consider the complexity of multiple-solution and u...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

A forest in which every component is path is caned a path forest. A family of path forests whose edge sets form a partition of the set of a graph G is called a path of G. The minimum number of forests in a path decomposition of a graph G is the path number of G and denoted by p( G). If we restrict the number of edges in each path to be at most x then we obtain a special decomposition. The minim...

We study the problem of constructing graphs from shortest path information (complete or partial). Consider graphs with labeled vertices and edges. Given a collection V of vertices and for each u 2 V a positive integer d(u), and a family F u = fF u;i : i < d(u)g of subsets of V construct a graph such that for each u and each link i of u, F u;i is the set of nodes having an optimal length path to...

The binding number of a graph was introduced by D.R.Woodall in 1973 [10] and is defined as the minimum of the ratios | (X)| /|X| taken over all non-empty subsets ofX of vertices inG such that (X) = V (G), where (X) = ∪v∈X (v) and (v) the set of all vertices adjacent to a vertex v in G. We obtain exact values of the binding number of middle graph of cycle, path, complete graph and complete bipar...