Circular Chromatic Index of Generalized Blanuša Snarks

Abstract

In his Master's thesis, Ján Mazák proved that the circular chromatic index of the type 1 generalized Blanuša snark $B^1_n$ equals $3+{2\over n}$. This result provided the first infinite set of values of the circular chromatic index of snarks. In this paper we show the type 2 generalized Blanuša snark $B^2_n$ has circular chromatic index $3+{1/\lfloor{1+3n/2}\rfloor}$. In particular, this proves that all numbers $3+1/n$ with $n\ge 2$ are realized as the circular chromatic index of a snark. For $n=1,2$ our proof is computer-assisted.

Asymptotic Lower Bounds on Circular Chromatic Index of Snarks

“…The class of generalised type 1 Blanuša snarks contains a snark on 8m + 2 vertices with circular chromatic index 3 + 2/3m for every positive integer m (see [13] for the proof). This shows that b (5,4) 8/3 = 2.6. Despite our effort we have not found any snark which would give a better upper bound on the constant b (5,4).…”

“…This shows that b (5,4) 8/3 = 2.6. Despite our effort we have not found any snark which would give a better upper bound on the constant b (5,4). We conjecture that b(5, 4) = 2.6.…”

“…The upper bound of 11/3 for bridgeless snarks was proved by Afshani et al [1]. The values of circular chromatic index have been determined for Isaac snarks [6], Goldberg snarks [4], a few classes of graphs that are trivial in some sense (for instance, see [16]), generalised Blanuša snarks [13,5], and the cubic graphs that have been constructed to show that any rational number from the interval (3, 10/3) is equal to the circular chromatic index of some cubic graph [11].…”

Circular chromatic index of type 1 Blanuša snarks

“…An analysis of the block A 2 in a manner similar to the one described in this article is essential for determining the circular chromatic index of type 2 Blanuša snarks. This has been recently done in [3] by using certain results and methods developed in the master's thesis [8] of the present author. The result of this article is also part of that thesis.…”

Circular edge-colorings of cubic graphs with girth six

“…This latter result was generalized to graphs with bounded maximum degree [7]. Moreover, the circular chromatic indices of several well-known classes of snarks have been determined [2,3,4,10].…”