Computing the circular chromatic number of a given planar graph is NP-complete, as it is already NP-complete to compute its chromatic number. In this note, we prove that the circular clique number of a planar graph, and therefore the circular chromatic number of a circular perfect graph, is computable in O(ne) time; outerplanar graphs are circular perfect.

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing. More precisely, let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-packing. We show that

Abstract We develop a theory of graph algebras over general fields. This is modelled after the developed by Freedman et al. (2007, J. Amer. Math. Soc. 20 37–51) for connection matrices, in study homomorphism functions real edge weight and positive vertex weight. introduce tensors properties. notion naturally generalizes concept matrices. It shown that counting perfect matchings, host other prop...

Let G be a directed graph. Its vertex-set will be denoted by X, and its arcs (or ‘directed edges’) are a subset of the Cartesian product X x X. A kernel of G is a subset S of X which is ‘stable’ (independent, i.e.: a vertex in S has no successor in S) and ‘absorbant’ (dominating, i.e. a vertex not in S has a successor in S). This concept has found many applications, for instance in cooperative ...

A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of G). A cubic graph G is called a Kotzig graph if G has a 3-edge-coloring such t...

Let H be a hexagonal system. The Z-transformation graph Z(H) is the graph where the vertices are the perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric difference is a hexagon of H (Z. Fu-ji et al., 1988). In this paper we prove that Z(H) has a Hamilton path if H is a catacondensed hexagonal system. A hexagonal system [ll], also called honeycom...