Circular choosability

Abstract

International audienceWe study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O(ch(G) + ln |V(G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f-choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of τ := sup {cch(G) : G is planar}, and we prove that 6<τ>8, thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density

On two questions about circular choosability

“…In Section 2, we show that the circular choosability for even cycles χ c,l (C 2k ) is equal to 2, answering a question of Zhu [8] and verifying a conjecture of Havet et al [2]. The circular choosability for odd cycles was computed by Zhu in [7], where he shows that χ c,l (C 2k+1 ) = 2 + 1 k .…”

“…The following important questions about circular choosability posed by Zhu [7] We refer the reader to [2,8] for a more comprehensive list of known results and open problems about circular choosability.…”

“…The circular choosability for odd cycles was computed by Zhu in [7], where he shows that χ c,l (C 2k+1 ) = 2 + 1 k . In Section 3, we compute the circular choosability for the complete bipartite graph K 2,4 . We show that χ c,l (K 2,4 ) = 2, while K 2,4 is not circular 2-choosable.…”

“…Let G = (V (G), E(G)) be a graph and let p and q be positive integers (following [2] we relax the requirement p ≥ The circular chromatic number is a refinement of the chromatic number (χ(G) − 1 < χ c (G) ≤ χ(G) for any graph G [4,6]) and as such might reflect more structural properties of the graph than the chromatic number. It has been extensively studied in recent years, see [8] for a survey of the subject.…”

Circular choosability via combinatorial Nullstellensatz

“…The circular list chromatic numbers are known for trees, cycles, complete graphs, odd wheels, etc. It is known that graphs of maximum degree k have circular list chromatic number at most k + 1 [11], the circular list chromatic number of planar graphs is at most 8 [3], outerplanar graphs G of girth at least 2n + 2 are shown to have χ c,l (G) ≤ 2 + 1/n [9]. Circular list coloring of graphs in which the list L(v) of permissible colors assigned to each vertex v is a circular consecutive set is studied in [4].…”