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...

A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs. Moreover, we ...

Let G be a cubic graph with a perfect matching. An edge e of G is a forcing edge if it is contained in a unique perfect matching M , and the perfect matching M is called uniquely forced. In this paper, we show that a 3-connected cubic graphwith a uniquely forced perfect matching is generated from K4 via Y → 1-operations, i.e., replacing a vertex by a triangle, and further a cubic graph with a u...

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 tiroes the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduc...

A graph is called t-perfect if its stable set polytope is defined by nonnegativity, edge and odd-cycle inequalities. We show that it can be decided in polynomial time whether a given claw-free graph is t-perfect.