Circular chromatic index of Cartesian products of graphs

Abstract

Abstract:The circular chromatic index of a graph G, written χ c (G), is the minimum r permitting a function f : E(G) → [0, r) such that 1 ≤ |f(e) − f(e )| ≤ r − 1 whenever e and e are incident. Let G = H C 2m+1 , where denotes Cartesian product and H is an (s − 2)-regular graph of odd order, with

Circular chromatic indices of even degree regular graphs

“…We shall use frequently the following lemma and its consequence proved in [18] (see Lemma 2.1 of [18]). Lemma 3.3.…”

“…The circular chromatic index of snarks has been studied in [1,[4][5][6]14], and the circular chromatic index of Cartesian product graphs has been studied in [18]. Nevertheless, the question as which real numbers are the circular chromatic indices of graphs remains largely open.…”

“…Thus we have χ ′ (G) − 1 < χ ′ c (G) ≤ χ ′ (G), and χ ′ c is a refinement of χ ′ . The circular chromatic indices of graphs have been studied in many papers [1,[3][4][5][6][8][9][10][12][13][14]16,18]. It is known that for a subcubic multigraph G, either χ ′ c (G) = 4 or χ ′ c (G) ≤ 11/3 [1]; subcubic graphs G of girth at least six have χ ′ c (G) ≤ 7/2 [10]; graphs G of large girth have χ ′ c (G) close to ∆(G) [9]; for any integers k ≥ 4, 1 ≤ a ≤ k/2, p ≥ 2a 2…”

DP‐coloring Cartesian products of graphs

“…In this paper all graphs are nonempty, finite, and simple unless otherwise noted. Generally speaking we follow West [24] for terminology and notation. We use to denote the set of all natural numbers.…”

Circular Chromatic Index of Generalized Blanuša Snarks

“…For example the circular chromatic index of the flower snarks is studied in [3] and the circular chromatic index of Goldberg snarks and twisted Goldberg snarks is studied in [2]. West and Zhu [8] study the circular chromatic index of Cartesian products of graphs, and toroidal grids in particular. It is convenient when studying edge-colourings, to allow semiedges in graphs, i.e.…”