(2 + ?)-Coloring of planar graphs with large odd-girth

Abstract

The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f ( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f ( ), then G is (2 + )-colorable. Note that the function f ( ) is independent of the graph G and → 0 if and only if f ( ) → ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd-girth of planar graphs. This lemma is expected to have applications in related problems.

List circular coloring of trees and cycles

“…, 2k}. The trees T used in [2] and [7] are paths, and the trees T used in [1,4,12] are star-like trees. The results obtained so far concerning the circular chromatic number of planar graphs of large girth is still far from a conjectured value [3,5,6].…”

“…Circular list coloring is also motivated by the need in the inductive proofs of circular coloring results. For example, the circular chromatic number of planar graphs of large odd girth is studied extensively in the literature [1,2,4,7,12]. A common feature of the proofs in these papers is that one needs to extend a (2k + 1, k)-coloring of a special subgraph G to the entire graph G, where G = G − T , where T is a tree.…”

Graphs with bounded tree-width and large odd-girth are almost bipartite

“…Thus, the high odd-girth requirement is not sufficient to ensure 3-colorability, even for graphs embedded on a fixed surface. Klostermeyer and Zhang [4], though, proved that the circular chromatic number of every planar graph of sufficiently high oddgirth is arbitrarily close to 2. In particular, the same is true for K 4 -minor free graphs, i.e.…”

Graph factors modulo k

“…In [14] it is shown precisely which flow values can be used in the (2 + )-flow conjecture. Prior to that, the (2 + )-flow conjecture had been verified first for planar graphs [6] and then for graphs on a fixed surface [16]. Apart from these results not much was known about the conjecture, as pointed out in [17] In this paper we reformulate the orientation result in the weak circular flow conjecture as a factor result for bipartite graphs and derive the special case mentioned in the Abstract.…”