Let x(G) and w(G) denote the chromatic number and clique number (maximum size of a clique) of a graph G. To avoid trivial cases, we always assume that w (G);?: 2. It is well known that interval graphs are perfect, in particular x( G)= w (G) for every interval graph G. In this paper we study the closeness of x and w for two well-known non-perfect relatives of interval graphs: multiple interval g...

We investigate two modifications of the well-known irregularity strength of graphs, namely, a total edge irregularity strength and a total vertex irregularity strength. Recently the bounds and precise values for some families of graphs concerning these parameters have been determined. In this paper, we determine the exact value of the total edge (vertex) irregularity strength for the disjoint u...

It has been known for some time that the Higman-Sims graph can be decomposed into the disjoint union of two Hoffman-Singleton graphs. In this paper we establish that the Higman-Sims graph can be edge decomposed into the disjoint union of 5 double-Petersen graphs, each on 20 vertices. It is shown that in fact this can be achieved in 36960 distinct ways. It is also shown that these different ways...

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (...

A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in R. A more general theory is developed for frameworks in R whose vertices are constrained to move on a two-dimensional smooth submanifold M. Furthermore, when M is a union of concentric spheres, or a union of parallel planes or a union of concentric...

A directed star forest is a forest all of whose components are stars with arcs emanating from the center to the leaves. The acircuitic directed star arboricity of an oriented graph G is the minimum number of edge-disjoint directed star forests whose union covers all edges of G and such that the union of two such forests is acircuitic. We show that graphs with maximum average degree less than 7 ...