Vertex stress related parameters for certain Kneser graphs

Let n and k be positive integers. The Kneser graph K 2n+k n is the graph with vertex set [2n+k] n and where two n-subsets A, B ∈ [2n+k] n are joined by an edge if A∩B = ∅. In this note we show that the diameter of the Kneser graph K 2n+k n is equal to n−1 k + 1.

arXiv (Cornell University), 2021

A set D of vertices of a graph G is a dissociation set if each vertex of D has at most one neighbor in D. The dissociation number of G, diss(G), is the cardinality of a maximum dissociation set in a graph G. In this paper we study dissociation in the well-known class of Kneser graphs K n,k. In particular, we establish that the dissociation number of Kneser graphs Kn,2 equals max {n − 1, 6}. We show that for any k ≥ 2, there exists n0 ∈ N such that diss(K n,k) = α(K n,k) for any n ≥ n0. We consider the case k = 3 in more details and prove that n0 = 8 in this case. Then we improve a trivial upper bound 2α(K n,k) for the dissociation number of Kneser graphs K n,k by using Katona's cyclic arrangement of integers from {1,. .. , n}. Finally we investigate the odd graphs, that is, the Kneser graphs with n = 2k + 1. We prove that diss(K 2k+1,k) = 2k k .

Journal of the Indonesian Mathematical Society, 2021

This paper introduces new parameters called induced vertex stress and total induced vertex stress in G, respectively. For graphs G and H, aspects of the maximum and minimum total induced vertex stress that can be obtained by 1-edge addition and 2-vertex merging are discussed.

Advances and Applications in Discrete Mathematics

Let G be a simple graph. G L is the Laplacian matrix of G and G a 1 p p PG Z Z ℾ 2 p p Z Z and , p p PG Z Z where p is a prime. We also find their girth, vertex connectivity and discuss planarity and Eulerian properties.

Discrete Mathematics & Theoretical Computer Science, 2022

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

arXiv: Combinatorics, 2020

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

Discrete Mathematics & Theoretical Computer Science, 2013

Graph Theory International audience A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely Kn:k with n≥k(k+1) / 2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group Sn acting on the k-subsets of 1,..., n. Then, we establish for which Kneser graphs Kn:k the determining number is equal to n-k, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4.

2021

A vertex cut S of a connected graph G is a subset of vertices of G whose deletion makes G disconnected. A super vertex cut S of a connected graph G is a subset of vertices of G whose deletion makes G disconnected and there is no isolated vertex in each component ofG−S. The super-connectivity of graph G is the size of the minimum super vertex cut of G. Let KG(n, k) be the Kneser graph whose vertices set are the k-subsets of {1, · · · , n}, where k is the number of labels of each vertex in G. We aim to show that the conjecture from Boruzanli and Gauci [1] on the super-connectivity of Kneser graph KG(n, k) is true when k = 3.

Discrete Mathematics, 2008

A local coloring of a graph G is a function c : V (G) → N having the property that for each set S ⊆ V (G) with 2 ≤ |S| ≤ 3, there exist vertices u, v ∈ S such that |c(u) − c(v)| ≥ m S , where m S is the number of edges of the induced subgraph S . The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ (c). The local chromatic number of G is χ (G) = min{χ (c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K (n, k) for some values of n and k is determined.

2019

Given a connected graph G and an integer 1 ≤ p ≤ ⌊|V(G)|/2⌋, a p-restricted edge-cut of G is any set of edges S ⊂ E(G), if any, such that G − S is not connected and each component of G − S has at least p vertices; and the p-restricted edge-connectivity of G, denoted λp(G), is the minimum cardinality of such a p-restricted edge-cut. When p-restricted edge-cuts exist, G is said to be super-λp if the deletion from G of any p-restricted edge-cut S of cardinality λp(G) yields a graph G − S that has at least one component with exactly p vertices. In this work, we prove that Kneser graphs K(n, k) are λp-connected for a wide range of values of p. Moreover, we obtain the values of λp(G) for all possible p and all n ≥ 5 when G = K ( n , 2 ) . Also, we discuss in which cases λp(G) attains its maximum possible value, and determine for which values of p graph G = K ( n , 2 ) is super-λp.