Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

Abstract

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs.We use an old topological result of Ky Fan [17] which generalizes the Borsuk-Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K t/2 , t/2 in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of t/2 + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases.As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

Local chromatic number and distinguishing the strength of topological obstructions

“…(For the definition and basic properties of the fractional chromatic number we refer to the books [35] and [18].) This motivated in [38] the study of the local chromatic number of graphs that have a large gap between their ordinary and fractional chromatic numbers. Basic examples of such graphs include Kneser graphs and Mycielski graphs (see [35]) and their variants, the so-called Schrijver graphs (see [30], [36]) and generalized Mycielski graphs (see [19], [30], [41], [42]).…”

“…Another common feature of these graphs is that their chromatic number is (or at least can be) determined by the topological method initiated by Lovász in [28]. In [38] it is proved that for all these graphs of chromatic number t one has ψ(G) ≥ t 2 + 1, and several cases are shown when this bound is tight. In all those cases, however, we have an odd t; in particular, the smallest chromatic number for which [38] gives some Schrijver graphs, say, with smaller local than ordinary chromatic number, is 5, in spite of the fact that the lower bound t 2 +1 is smaller than t already for t = 4.…”

“…In [38] it is proved that for all these graphs of chromatic number t one has ψ(G) ≥ t 2 + 1, and several cases are shown when this bound is tight. In all those cases, however, we have an odd t; in particular, the smallest chromatic number for which [38] gives some Schrijver graphs, say, with smaller local than ordinary chromatic number, is 5, in spite of the fact that the lower bound t 2 +1 is smaller than t already for t = 4. In this paper we show that whether t = 4 or 5 is optimal in the above sense depends on the particular topological method that gives the chromatic number of the graph.…”

“…We are able to say more on the last two bounds that were singled out by the following definition in [38]. Definition 2.5.…”

“…(For the formal definition of all these graphs, see, e.g., [30] or [38].) One way to show that these graphs are strongly topologically t-chromatic is to refer to another simplicial complex, the neighborhood complex N (G) of the graph G, introduced by Lovász in [28].…”