scholarly journals The Distinguishing Chromatic Number of Kneser Graphs

10.37236/3066 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Zhongyuan Che ◽  
Karen L. Collins
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Chromatic Number ◽  
Finite Abelian Group ◽  
Proper Coloring ◽  
Kneser Graph ◽  
Nontrivial Automorphism ◽  
Vertex Set ◽  
Kneser Graphs ◽  
Element Set

A labeling $f: V(G) \rightarrow \{1, 2, \ldots, d\}$ 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 $\chi_D(G)$, is the minimum $d$ such that $G$ has a proper $d$-distinguishing labeling. Let $\chi(G)$ be the chromatic number of $G$ and $D(G)$ be the distinguishing number of $G$. Clearly, $\chi_D(G) \ge \chi(G)$ and $\chi_D(G) \ge D(G)$. Collins, Hovey and Trenk have given a tight upper bound on $\chi_D(G)-\chi(G)$ 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, $S_n$, such that $\chi_D(G) - \chi(G) \le 1$. In particular, we prove that $\chi_D(K(n,2))=\chi(K(n,2))+1$ for $n\ge 5$. In addition, we show that $\chi_D(K(n,r))=\chi(K(n,r))$ for $n \ge 2r+1$ and $r\ge 3$.

scholarly journals On the Chromatic Number of Matching Kneser Graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 1-21
Author(s):  
Meysam Alishahi ◽  
Hajiabolhassan Hossein
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Large Family ◽  
Permutation Graphs ◽  
Kneser Graph ◽  
Ulam Theorem ◽  
Turán Number ◽  
Vertex Set ◽  
Kneser Graphs ◽  
Host Graph

AbstractIn 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.

scholarly journals Independence Complexes of Stable Kneser Graphs

10.37236/605 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Benjamin Braun
Keyword(s):  
Chromatic Number ◽  
Homotopy Type ◽  
Kneser Graph ◽  
Homotopy Types ◽  
Vertex Set ◽  
Interesting Object ◽  
Independence Complex ◽  
Kneser Graphs ◽  
Object Of Study ◽  
Critical Class

For integers $n\geq 1$, $k\geq 0$, the stable Kneser graph $SG_{n,k}$ (also called the Schrijver graph) has as vertex set the stable $n$-subsets of $[2n+k]$ and as edges disjoint pairs of $n$-subsets, where a stable $n$-subset is one that does not contain any $2$-subset of the form $\{i,i+1\}$ or $\{1,2n+k\}$. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number $k+2$. This article contains a study of the independence complexes of $SG_{n,k}$ for small values of $n$ and $k$. Our contributions are two-fold: first, we prove that the homotopy type of the independence complex of $SG_{2,k}$ is a wedge of spheres of dimension two. Second, we determine the homotopy types of the independence complexes of certain graphs related to $SG_{n,2}$.

scholarly journals Stability Results for Vertex Turán Problems in Kneser Graphs

10.37236/8130 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dániel Gerbner ◽  
Abhishek Methuku ◽  
Dániel T. Nagy ◽  
Balazs Patkos ◽  
Máté Vizer
Keyword(s):  
Chromatic Number ◽  
General Theorem ◽  
Kneser Graph ◽  
Turan Problems ◽  
Vertex Set ◽  
Kneser Graphs ◽  
Stability Results

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free, was recently determined by Alishahi and Taherkhani, whenever $n$ is large enough compared to $k$ and $F$. In this paper, we determine the second largest size of a vertex set $W \subseteq V$ such that $K(n,k)[W]$ is $F$-free, in the case when $F$ is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of $F$. 

scholarly journals Using Determining Sets to Distinguish Kneser Graphs

10.37236/938 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Debra L. Boutin
Keyword(s):  
Kneser Graph ◽  
Nontrivial Automorphism ◽  
Vertex Set ◽  
Determining Set ◽  
Kneser Graphs

This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels. A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertex set using $1, \ldots, d$ so that no nontrivial automorphism of $G$ preserves the labels. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. We prove that a graph is $d$-distinguishable if and only if it has a determining set that can be $(d-1)$-distinguished. We use this to prove that every Kneser graph $K_{n:k}$ with $n\geq 6$ and $k\geq 2$ is $2$-distinguishable.

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

scholarly journals Colorful Subhypergraphs in Kneser Hypergraphs

10.37236/3573 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Frédéric Meunier
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Uniform Hypergraph ◽  
Proper Coloring ◽  
Kneser Graph ◽  
Complete Bipartite Graphs ◽  
Definition Of ◽  
Ky Fan

Using a $\mathbb{Z}_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a uniform hypergraph).

scholarly journals On the $P_3$-Hull Number of Kneser Graphs

10.37236/9903 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Luciano N. Grippo ◽  
Adrián Pastine ◽  
Pablo Torres ◽  
Mario Valencia-Pabon ◽  
Juan C. Vera
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Lower And Upper Bounds ◽  
Kneser Graph ◽  
Graph Homomorphisms ◽  
The Family ◽  
Vertex Set ◽  
Kneser Graphs

This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph. In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n>2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.

On the b-Chromatic Sum of Mycielskian of Km, n, Kn and Cn

2020 ◽  
Vol 20 (02) ◽  
pp. 2050007
Author(s):  
P. C. LISNA ◽  
M. S. SUNITHA
Keyword(s):  
Image Processing ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Proper Coloring ◽  
Color Class ◽  
Vertex Set ◽  
Chromatic Sum ◽  
Complete Bipartite Graphs

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

scholarly journals Dynamics of Endomorphism of Finite Abelian Groups

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Zhao Jinxing ◽  
Nan Jizhu
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Dynamical Systems ◽  
Automorphism Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Point System ◽  
Finite Abelian Groups

We study the dynamics of endomorphisms on a finite abelian group. We obtain the automorphism group for these dynamical systems. We also give criteria and algorithms to determine whether it is a fixed point system.

scholarly journals Bounds on the Distinguishing Chromatic Number

10.37236/177 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Mark Hovey ◽  
Ann N. Trenk
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Automorphism Group ◽  
Chromatic Number ◽  
Automorphism Groups ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Minimum Number

Collins and Trenk define the distinguishing chromatic number $\chi_D(G)$ of a graph $G$ to be the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism of $G$ that preserves colors is the identity. They prove results about $\chi_D(G)$ based on the underlying graph $G$. In this paper we prove results that relate $\chi_D(G)$ to the automorphism group of $G$. We prove two upper bounds for $\chi_D(G)$ in terms of the chromatic number $\chi(G)$ and show that each result is tight: (1) if Aut$(G)$ is any finite group of order $p_1^{i_1} p_2^{i_2} \cdots p_k^{i_k}$ then $\chi_D(G) \le \chi(G) + i_1 + i_2 \cdots + i_k$, and (2) if Aut$(G)$ is a finite and abelian group written Aut$(G) = {\Bbb Z}_{p_{1}^{i_{1}}}\times \cdots \times {\Bbb Z}_{p_{k}^{i_{k}}}$ then we get the improved bound $\chi_D(G) \le \chi(G) + k$. In addition, we characterize automorphism groups of graphs with $\chi_D(G) = 2$ and discuss similar results for graphs with $\chi_D(G)=3$.

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