Circular edge-colorings of cubic graphs with girth six

Abstract

We show that the circular chromatic index of a (sub)cubic graph with odd-girth at least 7 is at most 7/2.

Avoiding 5-Circuits in 2-Factors of Cubic Graphs

“…On the other hand, circuits of length 5 pose main obstacles in several problems (e.g. [9,1]) and our knowledge on how to avoid them was very limited except of some well defined situations.…”

Circular chromatic indices of even degree regular 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…”

Circular Chromatic Indices of Regular Graphs

“…To verify that (7/2, 11/3) is indeed a gap of S or M, one needs to prove that there is no subcubic graph G with χ c (G) ∈ (7/2, 11/3). Recently, a progress has been made in this direction: in [8] it was proved that if a subcubic graph G has odd girth at least 7, then χ c (G) ≤ 7/2. Gaps of S (or M) are intervals that are disjoint from S (or M).…”