Circular Chromatic Indices of Regular Graphs

Abstract

The circular chromatic index of a graph G, written χc′(G), is the minimum r permitting a function c:E(G)→[0,r) such that 1≤c(e)−c(e′)≤r−1 whenever e and e′ are adjacent. It is known that for any ε∈(0,1/3), there is a 3‐regular simple graph G with χc′(G)=3+ε. This article proves the following results: Assume n≥5 is an odd integer. For any ε∈(0,1/4), there is an n‐regular simple graph G with χc′(G)=n+ε. For any ε∈(0,1/3), there is an n‐regular multigraph G with χc′(G)=n+ε.

Circular chromatic indices of even degree regular graphs

“…In [12], we generalized this result and proved that for any odd integer n, [n, n + 1/3] is an M-interval and [n, n + 1/4] is an S-interval. In this paper, we extend this result and prove that for any even integer n, [n, n + 1/3] is an M-interval and [n, n + 1/6] is an S-interval.…”

“…However, to prove this theorem, it suffices to consider the case that r = n + ϵ are rationals. Indeed, to prove Theorem 2.1, we shall construct, for each rational r in the specified range, a finite regular graph G with χ ′ c (G) = r. As in [13,12], the graph G is obtained by gluing up small building blocks, called monochromatic networks.…”

“…The same proof works for the case that n is even, with one step needs a small modification. For the proof of Theorem 2.2 in [12], we first construct a graph G with χ ′ c (G) = r, where each vertex of G has degree n, except that some vertices has degree 2. Moreover, there is a circular r-edge-colouring of G such that, for each of these degree 2 vertices v, the colours assigned to the two edges incident to v are distance 1 apart.…”

“…The proof technique is similar to that in [13,12]. The difference is in the construction of the small pieces (monochromatic networks) that are used in the construction of the desired graphs.…”

“…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…”