### Abstract

For any graph G=(V,E) and positive integer p, the exact distance-p graph G^{[♮p]} is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G^{[♮p]} is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G^{[♮p]} is bounded by the weak (2p−1)-colouring number of G. For even p, we prove that χ(G^{[♮p]}) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G^{[♮p]} when G is planar is improved. Similar lower bounds are given for K_{t}-minor free graphs.

Original language | English (US) |
---|---|

Journal | Journal of Combinatorial Theory. Series B |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

### Fingerprint

### Keywords

- Bounded expansion
- Chromatic number
- Exact distance graphs
- Generalised colouring numbers
- Planar graphs

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*. https://doi.org/10.1016/j.jctb.2018.05.007

**Chromatic numbers of exact distance graphs.** / van den Heuvel, Jan; Kierstead, Henry; Quiroz, Daniel A.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*. https://doi.org/10.1016/j.jctb.2018.05.007

}

TY - JOUR

T1 - Chromatic numbers of exact distance graphs

AU - van den Heuvel, Jan

AU - Kierstead, Henry

AU - Quiroz, Daniel A.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - For any graph G=(V,E) and positive integer p, the exact distance-p graph G[♮p] is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[♮p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[♮p] is bounded by the weak (2p−1)-colouring number of G. For even p, we prove that χ(G[♮p]) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G[♮p] when G is planar is improved. Similar lower bounds are given for Kt-minor free graphs.

AB - For any graph G=(V,E) and positive integer p, the exact distance-p graph G[♮p] is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[♮p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[♮p] is bounded by the weak (2p−1)-colouring number of G. For even p, we prove that χ(G[♮p]) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G[♮p] when G is planar is improved. Similar lower bounds are given for Kt-minor free graphs.

KW - Bounded expansion

KW - Chromatic number

KW - Exact distance graphs

KW - Generalised colouring numbers

KW - Planar graphs

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U2 - 10.1016/j.jctb.2018.05.007

DO - 10.1016/j.jctb.2018.05.007

M3 - Article

AN - SCOPUS:85047953697

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -