A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. A star (biclique) k-coloring of G is a k-coloring of G that contains no monochromatic maximal stars (bicliques). Similarly, for a list assignment L of G, a star (biclique) L-coloring is an L-coloring of G in which no maximal star (biclique) is monochromatic...

AgraphG is free (a, b)-choosable if for any vertex v with b colors assigned and for any list of colors of size a associated with each vertex u = v, the coloring can be completed by choosing for u a subset of b colors such that adjacent vertices are colored with disjoint color sets. In this note, a necessary and sufficient condition for a cycle to be free (a, b)-choosable is given. As a corollar...

Abstract This paper considers list circular colouring of graphs in which the colour list assigned to each vertex is an interval of a circle. The circular consecutive choosability chcc(G) of G is defined to be the least t such that for any circle S(r) of length r ≥ χc(G), if each vertex x of G is assigned an interval L(x) of S(r) of length t, then there is a circular r-colouring f of G such that...

A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. A star (biclique) k-coloring of G is a k-coloring of G that contains no monochromatic maximal stars (bicliques). Similarly, for a list assignment L of G, a star (biclique) L-coloring is an L-coloring of G in which no maximal star (biclique) is monochromatic...

A total-weighting of a graph G = (V,E) is a mapping f which assigns to each element y ∈ V ∪ E a real number f(y) as the weight of y. A total-weighting f of G is proper if the colouring φf of the vertices of G defined as φf (v) = f(v) + ∑ e∋v f(e) is a proper colouring of G, i.e., φf (v) ̸= φf (u) for any edge uv. For positive integers k and k′, a graph G is called (k, k′)-total-weight-choosable ...

Let G be a planar graph with no two 3-cycles sharing an edge. We show that if ∆(G) ≥ 9, then χ′l(G) = ∆(G) and χ ′′ l (G) = ∆(G) + 1. We also show that if ∆(G) ≥ 6, then χ ′ l(G) ≤ ∆(G) + 1 and if ∆(G) ≥ 7, then χ′′ l (G) ≤ ∆(G) + 2. All of these results extend to graphs in the projective plane and when ∆(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-c...