The local chromatic number of a graph was introduced in [13]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the stable Kneser (or Schrijver) graphs; Mycielski graphs, and their generalizations; and...

A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. The minimum size of a label class in such a labeling of G is called the cost of 2-distinguishing and is denoted by ρ(G). This paper shows that ρ(K2m−1:2m−1−1) = m+1 – the only result so far on the cost of 2-distinguishing Kneser graphs. The...

Let r , k be positive integers, s(< r), a nonnegative integer, and n=2r−s+k. The set of r-subsets of [n]={1, 2, . . . , n} is denoted by [n]r . The generalized Kneser graphK(n, r, s) is the graph whose vertex-set is [n]r where two r-subsets A and B are joined by an edge if |A ∩ B| s. This note determines the diameter of generalized Kneser graphs. More precisely, the diameter of K(n, r, s) is eq...

The cutoff phenomenon for an ergodic Markov chain describes a sharp transition in the convergence to its stationary distribution, over a negligible period of time, known as cutoff window. We study the cutoff phenomenon for simple random walks on Kneser graphs, which is a family of ergodic Markov chains. Given two integers n and k, the Kneser graph K(2n+ k, n) is defined as the graph with vertex...

For integers n ≥ 1, k ≥ 0, the stable Kneser graph SGn,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 de...

A set of vertices S in a graph G is a resolving set for G if, for any two vertices u,v, there exists x ∈ S such that the distances d(u,x) 6= d(v,x). In this paper, we consider the Johnson graphs J(n,k) and Kneser graphs K(n,k), and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used...

We present some lower and upper bounds on the length of the maximum induced paths and cycles in Kneser graphs. The Kneser graph K(n, r) is the graph whose vertex set is the family of all r-element subsets of { 1, 2 . . . . . n}, and a pair of vertices forms an edge, ff the corresponding subsets are disjoint. Kneser graphs are often used in order to express some properties of a family of sets of...

The determining number of a graph $G = (V,E)$ is the minimum cardinality set $S\subseteq V$ such that pointwise stabilizer $S$ under action $Aut(G)$ trivial. In this paper, we provide some improved upper and lower bounds on Kneser graphs. Moreover, exact value for subfamilies

The problem of deciding whether an arbitrary graph G has a homomorphism into a given graph H has been widely studied and has turned out to be very difficult. Hell and Nešetril proved that the decision problem is NP-complete unless H is bipartite. We consider a restricted problem where G is an arbitrary triangle-free hexagonal graph and H is a Kneser graph or its induced subgraph. We give an exp...