A β-perfect graph is a simple graph G such that χ(G′) = β(G′) for every induced subgraph G′ of G, where χ(G′) is the chromatic number of G′, and β(G′) is defined as the maximum over all induced subgraphs H of G′ of the minimum vertex degree in H plus 1 (i.e., δ(H) + 1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily). The main purpose of this ...

The matching graph M(G) of a graph G has a vertex set of all perfect matchings of G, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We prove that the matching graph M(Qd) of the d-dimensional hypercube is bipartite and connected for d ≥ 4. This proves Kreweras’ conjecture [2] that the graph Md is connected, where Md is obtai...

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H , we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let δOre(H,n) be the smallest number k such that every graph G whose order n is di...

A one-factorization of a regular graph G is perfect if the union of any two one-factors is a Hamiltonian cycle. A graph G is said to be P1F if it possess a perfect one-factorization. We prove that G is a P1F cubic graph if and only if L(G) is a P1F quartic graph. Moreover, we give some necessary conditions for the existence of a P1F planar graph.

A total perfect code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is adjacent to exactly one vertex in the subset. We prove that the tensor product of any number of simple graphs has a total perfect code if and only if each factor has a total perfect code.

A perfect r-code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is within distance r of exactly one vertex in the subset. We prove that the n-fold strong product of simple graphs has a perfect r-code if and only if each factor has a perfect r-code.

A new algorithm for generating order preserving minimal perfect hash functions is presented. The algorithm is probabilistic, involving generation of random graphs. It uses expected linear time and requires a linear number words to represent the hash function, and thus is optimal up to constant factors. It runs very fast in practice.

A main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grötschel, Lovász and Schrijver 1981). Circular-perfect graphs form a well-studied superclass of perfect graphs. We extend the above result for perfect graphs by showing that clique and chromatic number of a circularperfect graph are computable in polynomial time...

The chromatic gap is the difference between the chromatic number and the clique number of a graph. Here we investigate gap(n), the maximum chromatic gap over graphs on n vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey theory and matching theory leads to a simple and (almost) exact formula for gap(...

We still do not know how to construct the “most general” perfect graph, not even the most general three-colourable perfect graph. But constructing all perfect graphs with no even pairs seems easier, and here we make a start on it; we construct all three-connected three-colourable perfect graphs without even pairs and without clique cutsets. They are all either line graphs of bipartite graphs, o...