### 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 language | English (US) |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 20 |

Issue number | 1 |

State | Published - Jan 29 2013 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Electronic Journal of Combinatorics*,

*20*(1).

}

*Electronic Journal of Combinatorics*, vol. 20, no. 1.

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

Research output: Contribution to journal › Article

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.

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M3 - Article

AN - SCOPUS:84873367543

VL - 20

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -