The distinguishing chromatic number of kneser graphs

Zhongyuan Che, Karen L. Collins

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

A labeling f: V (G) → {1, 2,..., d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by xD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let x(G) be the chromatic number of G and D(G) be the distinguishing number of G. Clearly, xD(G) > x(G) and xD(G) > D(G). Collins, Hovey and Trenk [6] have given a tight upper bound on xD(G) - x(G) in terms of the order of the automorphism group of G, improved when the automorphism group of G is a finite abelian group. The Kneser graph K(n; r) is a graph whose vertices are the r-subsets of an n element set, and two vertices of K(n; r) are adjacent if their corresponding two r-subsets are disjoint. In this paper, we provide a class of graphs G, namely Kneser graphs K(n; r), whose automorphism group is the symmetric group, Sn, such that xD(G) - x(G) 6 1. In particular, we prove that xD(K(n, 2)) = x(K(n, 2)) + 1 for n > 5. In addition, we show that xD(K(n, r)) = x(K(n, r)) for n ≥ 2r + 1 and r ≥ 3.

Original languageEnglish (US)
JournalElectronic Journal of Combinatorics
Volume20
Issue number1
StatePublished - Jan 29 2013

Fingerprint

Kneser Graph
Chromatic number
Automorphism Group
Labeling
Coloring
Graph in graph theory
Vertex of a graph
Labels
Subset
Finite Abelian Groups
Symmetric group
Automorphism
Colouring
Disjoint
Adjacent
Upper bound

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Theory and Mathematics

Cite this

@article{dd2104b5694d4c9cb56484af92acbc86,
title = "The distinguishing chromatic number of kneser graphs",
abstract = "A labeling f: V (G) → {1, 2,..., d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by xD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let x(G) be the chromatic number of G and D(G) be the distinguishing number of G. Clearly, xD(G) > x(G) and xD(G) > D(G). Collins, Hovey and Trenk [6] have given a tight upper bound on xD(G) - x(G) in terms of the order of the automorphism group of G, improved when the automorphism group of G is a finite abelian group. The Kneser graph K(n; r) is a graph whose vertices are the r-subsets of an n element set, and two vertices of K(n; r) are adjacent if their corresponding two r-subsets are disjoint. In this paper, we provide a class of graphs G, namely Kneser graphs K(n; r), whose automorphism group is the symmetric group, Sn, such that xD(G) - x(G) 6 1. In particular, we prove that xD(K(n, 2)) = x(K(n, 2)) + 1 for n > 5. In addition, we show that xD(K(n, r)) = x(K(n, r)) for n ≥ 2r + 1 and r ≥ 3.",
author = "Zhongyuan Che and Collins, {Karen L.}",
year = "2013",
month = "1",
day = "29",
language = "English (US)",
volume = "20",
journal = "Electronic Journal of Combinatorics",
issn = "1077-8926",
publisher = "Electronic Journal of Combinatorics",
number = "1",

}

The distinguishing chromatic number of kneser graphs. / Che, Zhongyuan; Collins, Karen L.

In: Electronic Journal of Combinatorics, Vol. 20, No. 1, 29.01.2013.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The distinguishing chromatic number of kneser graphs

AU - Che, Zhongyuan

AU - Collins, Karen L.

PY - 2013/1/29

Y1 - 2013/1/29

N2 - A labeling f: V (G) → {1, 2,..., d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by xD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let x(G) be the chromatic number of G and D(G) be the distinguishing number of G. Clearly, xD(G) > x(G) and xD(G) > D(G). Collins, Hovey and Trenk [6] have given a tight upper bound on xD(G) - x(G) in terms of the order of the automorphism group of G, improved when the automorphism group of G is a finite abelian group. The Kneser graph K(n; r) is a graph whose vertices are the r-subsets of an n element set, and two vertices of K(n; r) are adjacent if their corresponding two r-subsets are disjoint. In this paper, we provide a class of graphs G, namely Kneser graphs K(n; r), whose automorphism group is the symmetric group, Sn, such that xD(G) - x(G) 6 1. In particular, we prove that xD(K(n, 2)) = x(K(n, 2)) + 1 for n > 5. In addition, we show that xD(K(n, r)) = x(K(n, r)) for n ≥ 2r + 1 and r ≥ 3.

AB - A labeling f: V (G) → {1, 2,..., d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by xD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let x(G) be the chromatic number of G and D(G) be the distinguishing number of G. Clearly, xD(G) > x(G) and xD(G) > D(G). Collins, Hovey and Trenk [6] have given a tight upper bound on xD(G) - x(G) in terms of the order of the automorphism group of G, improved when the automorphism group of G is a finite abelian group. The Kneser graph K(n; r) is a graph whose vertices are the r-subsets of an n element set, and two vertices of K(n; r) are adjacent if their corresponding two r-subsets are disjoint. In this paper, we provide a class of graphs G, namely Kneser graphs K(n; r), whose automorphism group is the symmetric group, Sn, such that xD(G) - x(G) 6 1. In particular, we prove that xD(K(n, 2)) = x(K(n, 2)) + 1 for n > 5. In addition, we show that xD(K(n, r)) = x(K(n, r)) for n ≥ 2r + 1 and r ≥ 3.

UR - http://www.scopus.com/inward/record.url?scp=84873367543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84873367543&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84873367543

VL - 20

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -