A graph G is said to be f -choosable if there exists a proper coloring from every assignment of lists of colors to the vertices of G where the list sizes are given by f . The sum choice number of G is the minimum ∑ v∈V (G) f(v) over all f such that G is f -choosable. Here we determine the sum choice of the cartesian product P3 Pn to be 8n − 3 − bn/3c. The techniques used here have applicability...

A solution to a problem of Erdős, Rubin and Taylor is obtained by showing that if a graph G is (a : b)-choosable, and c/d > a/b, then G is not necessarily (c : d)-choosable. Applying probabilistic methods, an upper bound for the kth choice number of a graph is given. We also prove that a directed graph with maximum outdegree d and no odd directed cycle is (k(d + 1) : k)-choosable for every k ≥ ...

A proper vertex coloring of a simple graph is k-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than k. A graph is k-forested qchoosable if for a given list of q colors associated with each vertex v, there exists a k-forested coloring of G such that each vertex receives a color from its own list. In this paper, we prove that the k-fore...

Ohba has conjectured that if G is a k-chromatic graphwith at most 2k+1 vertices, then the list chromatic number or choosability ch(G) of G is equal to its chromatic number χ(G), which is k. It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete...

We study vertex colorings of the square G of an outerplanar graph G. We find the optimal bound of the inductiveness, chromatic number and the clique number of G as a function of the maximum degree ∆ of G for all ∆ ∈ N. As a bonus, we obtain the optimal bound of the choosability (or the list-chromatic number) of G when ∆ ≥ 7. In the case of chordal outerplanar graphs, we classify exactly which g...