The main result of this paper completely settles Bermond's conjecture for bipartite graphs of odd degree by proving that if G is a bipartite (2k+1)-regular graph that is Hamilton decomposable, then the line graph, L(G), of G is also Hamilton decomposable. A similar result is obtained for 5-regular graphs, thus providing further evidence to support Bermond's conjecture.

A graph G is ramdomly H–decomposable if every family of edge disjoint subgraphs of G, each subgraph isomorphic to H , can be extended to an H–decomposition of G. Let Pk denote a path of length k. In this paper we characterize all ramdomly P3– decomposable graphs and ramdomly P4–decomposable graphs. We also characterize all ramdomly Pk–decomposable graphs of size 2k where all vertices have even ...

the crossing number of a graph  is the minimum number of edge crossings over all possible drawings of  in a plane. the crossing number is an important measure of the non-planarity of a graph, with applications in discrete and computational geometry and vlsi circuit design. in this paper we introduce vertex centered crossing number and study the same for maximal planar graph.

The eccentricity of a vertex $v$ is the maximum distance between $v$ and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum ofthe distances between all pairs of peripheral vertices of $G.$ In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperti...

In this paper, the composite order Cayley graph Cay(G, S) is introduced, where G is a group and S is the set of all composite order elements of G. Some graph parameters such as diameter, girth, clique number, independence number, vertex chromatic number and domination number are calculated for the composite order Cayley graph of some certain groups. Moreover, the planarity of composite order Ca...

Abstract The junction-tree representation provides an attractive structural property for organising a decomposable graph . In this study, we present two novel stochastic algorithms, referred to as the expander and collapser , sequential sampling of junction trees graphs. We show that recursive application expander, which expands incrementally underlying with one vertex at time, has full support...

An abstract convexity space on a connected hypergraph H with vertex set V (H) is a family C of subsets of V (H) (to be called the convex sets of H) such that: (i) C contains the empty set and V (H), (ii) C is closed under intersection, and (iii) every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by ...

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...