Fractionally colouring total graphs

Kilakos, K.; Reed, Bruce

doi:10.1007/bf01303515

Cited by 23 publications

(12 citation statements)

References 5 publications

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“…Note that if true, this conjecture is tight as every such graph requires at least ∆+1 colours and there are some graphs such as K ∆+1 , ∆ odd, which require ∆+2 colours. Kilakos and Reed [15] have shown that the fractional total chromatic number is at most ∆ + 2. For more information on total colouring, see the recent book by Yap [23].…”

Section: Introductionmentioning

confidence: 99%

A Bound on the Total Chromatic Number

Molloy

Reed

1998

Combinatorica

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Section: Introductionmentioning

confidence: 99%

A Bound on the Total Chromatic Number

Molloy

Reed

1998

Combinatorica

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“…It was solved for ∆(G) = 3 by Rosenfeld [36] and Vijayaditya [50] (it is trivial for ∆(G) ≤ 2) and for ∆(G) = 4 and 5 by Kostochka [25,26,27]. The fractional chromatic number χ f (T (G)) is proven to be at most ∆(G) + 2 for any value of ∆(G) by Kilakos and Reed [23].…”

Section: On the Behzad-vizing Conjecturementioning

confidence: 99%

“…[37].) Examples include the Behzad-Vizing conjecture [23], the Erdős-Faber-Lovász conjecture [19], Hedetniemi's conjecture [46], a relaxed version of Hadwiger's conjecture [34], as well as a similarly relaxed version of the so-called odd Hadwiger conjecture [21]. (In some of these cases the proven fractional version has an approximative form, nevertheless, it is a statement not known to hold for the chromatic number.…”

Section: Introductionmentioning

confidence: 99%

On topological relaxations of chromatic conjectures

Simonyi

Zsbán

2010

European Journal of Combinatorics

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There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number

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“…. Circular chromatic number and fractional chromatic number and their variations have been extensively studied in the last two decades [6][7][8][9][10][11]. More results on the circular chromatic number and fractional chromatic number can be found in the book [12] and the survey papers [13,14].…”

Section: Introductionmentioning

confidence: 99%

Multiple Circular Colouring as a Model for Scheduling

Zhou¹

2013

OJDM

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In this article we propose a new model for scheduling periodic tasks. The model is based on a variation of the circular chromatic number, called the multiple circular colouring of the conflict graph. We show that for a large class of graphs, this new model will provide better solutions than the original circular chromatic number. At the same time, it allows us to avoid the difficulty of implementation when the fractional chromatic number is used.

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Fractionally colouring total graphs

Cited by 23 publications

References 5 publications

A Bound on the Total Chromatic Number

A Bound on the Total Chromatic Number

On topological relaxations of chromatic conjectures

Multiple Circular Colouring as a Model for Scheduling

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