The Choice Number Versus the Chromatic Number for Graphs Embeddable on Orientable Surfaces

We show that for loopless $6$-regular triangulations on the torus the gap between the choice number and chromatic number is at most $2$. We also show that the largest gap for graphs embeddable in an orientable surface of genus $g$ is of the order $\Theta(\sqrt{g})$, and moreover for graphs with chromatic number of the order $o(\sqrt{g}/\log_{2}(g))$ the largest gap is of the order $o(\sqrt{g})$.

Edge (m,k)-choosability of graphs

Let [Formula: see text] be a simple, connected graph of order [Formula: see text] and size [Formula: see text] Then, [Formula: see text] is said to be edge [Formula: see text]-choosable, if there exists a collection of subsets of the edge set, [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are incident. This paper initiates a study on edge [Formula: see text]-choosability of certain fundamental classes of graphs and determines the maximum value of [Formula: see text] for which the given graph [Formula: see text] is edge [Formula: see text]-choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph.

Planar graphs without intersecting 5-cycles are signed-4-choosable

A signed graph is a pair [Formula: see text], where [Formula: see text] is a graph and [Formula: see text] is a signature of [Formula: see text]. A set [Formula: see text] of integers is symmetric if [Formula: see text] implies that [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an [Formula: see text]-coloring of a signed graph [Formula: see text] is a coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text] and [Formula: see text] for every edge [Formula: see text]. The signed choice number [Formula: see text] of a graph [Formula: see text] is defined to be the minimum integer [Formula: see text] such that for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] and for any signature [Formula: see text] on [Formula: see text], there is a proper [Formula: see text]-coloring of [Formula: see text]. List signed coloring is a generalization of list coloring. However, the difference between signed choice number and choice number can be arbitrarily large. Hu and Wu [Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792] showed that every planar graph without intersecting 5-cycles is 4-choosable. In this paper, we prove that [Formula: see text] if [Formula: see text] is a planar graph without intersecting 5-cycles, which extends the main result of [D. Hu and J. Wu, Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792].

2-distance choice number of planar graphs with maximal degree no more than 4

Regarding the 2-[Formula: see text] coloring of planar graphs, in 1977, Wegner conjectured that for a graph [Formula: see text]: (1) [Formula: see text] if [Formula: see text]. (2) [Formula: see text] if [Formula: see text]. (3) [Formula: see text] if [Formula: see text]. In this paper, we proved that for every planar graph with maximum degree [Formula: see text]: (1) [Formula: see text] if [Formula: see text]. (2) [Formula: see text] if [Formula: see text].

Two results on K-(2,1)-total choosability of planar graphs

The [Formula: see text]-total choice number of [Formula: see text], denoted by [Formula: see text], is the minimum [Formula: see text] such that [Formula: see text] is [Formula: see text]-[Formula: see text]-total choosable. It was proved in [Y. Yu, X. Zhang and G. Z. Liu, List (d,1)-total labeling of graphs embedded in surfaces, Oper. Res. Trans. 15(3) (2011) 29–37.] that [Formula: see text] if [Formula: see text] is a graph embedded in surface with Euler characteristic [Formula: see text] and [Formula: see text] big enough. In this paper, we prove that: (i) if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text]-cycle is not adjacent to [Formula: see text]-cycle, [Formula: see text], then [Formula: see text]; (ii) if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text]-cycle is not adjacent to [Formula: see text]-cycle, where [Formula: see text], then [Formula: see text].