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On the Chromatic Number of Matching Kneser Graphs

Part of: Graph theory

Published online by Cambridge University Press:  12 September 2019

Meysam Alishahi  and
Hajiabolhassan Hossein
Meysam Alishahi
Affiliation:
Faculty of Mathematical Sciences, Shahrood University of Technology, Iran
Hajiabolhassan Hossein*
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Lyngby, Denmark Department of Mathematical Sciences, Shahid Beheshti University, P.O. Box 19839-69411, Tehran, Iran
*
*Corresponding author. Email: [email protected]
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Abstract

In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 1 , January 2020 , pp. 1 - 21
Copyright
© Cambridge University Press 2019 

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