Tight relation between the circular chromatic number and the girth of series-parallel graphs

We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k + 1 admits a homomorphism to the projective hypercube of dimension 2k. This supports a conjecture of the third author which generalizes the four-color theorem and relates to several outstanding conjectures such as Seymour's conjecture on edge-colorings of planar graphs. Strengthening this result, we show that the Kneser graph K(2k + 1, k) satisfies the conditions, thus implying that every K4-minor free graph of odd-girth 2k + 1 has fractional chromatic number exactly 2 + 1 k . Knowing that a smallest bound of odd-girth 2k + 1 must have at least k+2 2 vertices, we build nearly optimal bounds of order 4k 2 . Furthermore, we conjecture that the suprema of the fractional and circular chromatic numbers for K4-minor-free graphs of odd-girth 2k + 1 are achieved by a same bound of odd-girth 2k + 1. If true, this improves, in the homomorphism order, earlier tight results on the circular chromatic number of K4-minor-free graphs. We support our conjecture by proving it for the first few cases. Finally, as an application of our work, and after noting that Seymour provided a formula for calculating the edge-chromatic number of K4-minor-free multigraphs, we show that stronger results can be obtained in the case of K4-minor-free regular multigraphs. *

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“…Theorem 10 (Pan and Zhu [37]). For any > 0, there is a graph of SP 7 with circular chromatic number at least 5/2 − .…”

Section: Circular Chromatic Numbermentioning

confidence: 99%

Homomorphism bounds and edge-colourings of K4-minor-free graphs

Beaudou

Foucaud

Naserasr

2017

Journal of Combinatorial Theory, Series B

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“…Secondly, based on a good understanding of the relationship between the circular chromatic number and the girth (as well as the odd girth) of series-parallel graphs [1,6,11,12], we can show that the analogue of Conjecture 4.1 for series-parallel graphs is true. (Series-parallel graphs are graphs obtained from K 2 by repeatedly applying two operations: subdividing an edge and duplicating an edge.)…”

Section: A Remark On the Relation Between G(k) And F (K)mentioning

confidence: 97%

Circular Chromatic Number of Planar Graphs of Large Odd Girth

Zhu¹

2001

Electron. J. Combin.

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It was conjectured by Jaeger that $4k$-edge connected graphs admit a $(2k+1, k)$-flow. The restriction of this conjecture to planar graphs is equivalent to the statement that planar graphs of girth at least $4k$ have circular chromatic number at most $2+ {{1}\over {k}}$. Even this restricted version of Jaeger's conjecture is largely open. The $k=1$ case is the well-known Grötzsch 3-colour theorem. This paper proves that for $k \geq 2$, planar graphs of odd girth at least $8k-3$ have circular chromatic number at most $2+{{1}\over {k}}$.

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“…then the two terminals are denoted by x and y (respectively x 0 and y 0 , x 00 and y 00 , etc.). The circular chromatic number of seriesparallel graphs was considered in [2,3,8,9].…”

Section: Two Terminal Series-parallel Graphsmentioning

confidence: 99%