Kneser Graphs Research Articles

Clique number of Xor products of Kneser graphs

In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in “A general 2-part Erdős-Ko-Rado theorem” by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xor product of two isomorphic KG(N,k) Kneser graphs. Denote this number with f(k,N). We give lower and upper bounds on f(k,N), and we solve the problem up to a constant deviation depending only on k, and find the exact value for f(2,N) if N is large enough. Also we compute that f(k,k2) is asymptotically equivalent to k2.

Some Results on Dominating Induced Matchings

Let G be a graph, a dominating induced matching (DIM) of G is an induced matching that dominates every edge of G. In this paper we show that if a graph G has a DIM, then \(\chi (G) \le 3\). Also, it is shown that if G is a connected graph whose all edges can be partitioned into DIM, then G is either a regular graph or a biregular graph and indeed we characterize all graphs whose edge set can be partitioned into DIM. Also, we prove that if G is an r-regular graph of order n whose edges can be partitioned into DIM, then n is divisible by \(\left( {\begin{array}{c}2r - 1\\ r - 1\end{array}}\right) \) and \(n = \left( {\begin{array}{c}2r - 1\\ r-1\end{array}}\right) \) if and only if G is the Kneser graph with parameters \(r-1\), \(2r-1\).

Strong edge colorings of graphs and the covers of Kneser graphs

AbstractA proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a ‐regular graph at least colors are needed. We show that a ‐regular graph admits a strong edge coloring with colors if and only if it covers the Kneser graph . In particular, a cubic graph is strongly 5‐edge‐colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. is false.

Studying the multichromatic number of the almost s-stable Kneser graphs

In the early 1970’s Gilbert introduced n-tuple colorings of graphs motivated by practical problems. After this Saul Stahl studied the properties of these colorings and formulated the conjecture on the multichromatic number of the Kneser graphs. Motivated by Stahl’s conjecture we will investigate the multichromatic number of the almost s-stable Kneser graphs.

Distance strings of the vertices of certain graphs

The notion of the distance string of a vertex $v_i \in V(G)$ which is denoted by, $\tau(v_i)$ is introduced. Distance strings permit a new approach to determining the induced vertex stress, the total induced vertex stress and total vertex stress (sum of vertex stress over all vertices) of a graph. A seemingly under-studied topic i.e. the eccentricity of a vertex of a bipartite Kneser graph $BK(n,k)$, $n \geq 2k + 1$ has been furthered. A surprisingly simple result was established, namely for $k \geq 2$, $diam(BK(n,k)) = 5$ if $n = 2k + 1$ and $diam(BK(n,k)) = 3$ if $n \neq 2k + 1$.

Vertex stress related parameters for certain Kneser graphs

Abstract This paper presents results for some vertex stress related parameters in respect of specific subfamilies of Kneser graphs. Kneser graphs for which diam(KG(n, k)) = 2 and k ≥ 2 are considered. The note establishes the foundation for researching similar results for Kneser graphs for which diam(KG(n, k)) ≥ 3. In addition some important vertex stress related properties are stated. Finally some results for specific bipartite Kneser graphs i.e. BK(n, 1), n ≥ 3 will be presented. In the conclusion some worthy research avenues are proposed.

On the chromatic number of two generalized Kneser graphs

We determine the chromatic number of some graphs of flags in buildings of type A4, namely of the Kneser graphs of flags of type {2,4} in the vector spaces GF(q)5 for q≥3, and of the Kneser graph of flags of type {2,3} in the vector spaces GF(q)5 for large q.

On total and edge coloring some Kneser graphs

On total and edge coloring some Kneser graphs

Achromatic numbers of Kneser graphs

Complete vertex colorings have the property that any two color classes have at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers come from complete colorings, with some additional requirement. In this paper, we estimate these numbers in the Kneser graph K ( n , k ) for some values of n and k . We give the exact value of the achromatic number of K ( n , 2) .

On the $P_3$-Hull Number of 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.

Certain homological invariants of bipartite kneser graphs

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of these graphs in terms of combinatorial data.

The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $\mathrm{PG}(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\geqslant 43$ we determine the largest independent set of $\Gamma$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\geqslant 47$, this information is then used to show that $\Gamma$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.

The toughness of Kneser graphs

The toughnesst(G) of a graph G is a measure of its connectivity that is closely related to Hamiltonicity. Xiaofeng Gu, confirming a longstanding conjecture of Brouwer, recently proved the lower bound t(G)≥ℓ/λ−1 on the toughness of any connected ℓ-regular graph, where λ is the largest nontrivial absolute eigenvalue of the adjacency matrix. Brouwer had also observed that many families of graphs (in particular, those achieving equality in the Hoffman ratio bound for the independence number) have toughness exactly ℓ/λ. Cioabă and Wong confirmed Brouwer's observation for several families of graphs, including Kneser graphs K(n,2) and their complements, with the exception of the Petersen graph K(5,2). In this paper, we extend these results and determine the toughness of Kneser graphs K(n,k) when k∈{3,4} and n≥2k+1 as well as for k≥5 and sufficiently large n (in terms of k). In all these cases, the toughness is attained by the complement of a maximum independent set and we conjecture that this is the case for any k≥5 and n≥2k+1.

Colorings of complements of line graphs

AbstractOur purpose is to show that complements of line graphs (of graphs) enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph that have a chromatic number equal to its chromatic number, namely . In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the NP‐hardness of deciding the equality between the chromatic number and any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this article are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results.

Short proof that Kneser graphs are Hamiltonian for n ⩾ 4k

Let n⩾k⩾1 be integers and define [n]={1,…,n}. For a set A let A(k) be the set of all k-element subsets of A. The Kneser graph K(n,k) is the graph with vertex set V=[n](k) and edge set E={{x,y}∈V(2):x∩y=∅}. Chen proved that for n⩾3k, Kneser graphs are Hamiltonian and later improved this to n⩾2.62k+1. Furthermore, Chen and Füredi gave a short proof that if k|n, Kneser graphs are Hamiltonian for n⩾3k. In this note, we present a short proof that does not need the divisibility condition, i.e., we give a short proof that K(n,k) is Hamiltonian for n⩾4k.

Automorphism groups of some families of bipartite graphs

This paper discusses the automorphism group of a class of weakly semiregular bipartite graphs and its subclass called WSB END graphs. It also tries to analyse the automorphism group of the SM sum graphs and SM balancing graphs. These graphs are weakly semiregular bipartite graphs too. The SM sum graphs are particular cases of bipartite Kneser graphs. The bipartite Kneser type graphs are defined on n -sets for a fixed positive integer n . The automorphism groups of the bipartite Kneser type graphs are related to that of weakly semiregular bipartite graphs. Weakly semiregular bipartite graphs in which the neighbourhoods of the vertices in the SD part having the same degree sequence, possess non trivial automorphisms. The automorphism groups of SM sum graphs are isomorphic to the symmetric groups. The relationship between the automorphism groups of SM balancing graphs and symmetric groups are established here. It has been observed by using the well known algorithm Nauty, that the size of automorphism groups of SM balancing graphs are prodigious. Every weakly semiregular bipartite graphs with k-NSD subparts has a matching which saturates the smaller partition.

On the neighborhood complex of [formula omitted]-stable Kneser graphs

In 2002, Björner and de Longueville showed the neighborhood complex of the 2-stable Kneser graph KG(n,k)2−stab has the same homotopy type as the (n−2k)-sphere. A short time ago, an analogous result about the homotopy type of the neighborhood complex of almost s-stable Kneser graph has been announced by the second author. Combining this result with the famous Lovász’s topological lower bound on the chromatic number of graphs yielded a new way for determining the chromatic number of these graphs which was determined a bit earlier by Chen.In this paper we present a common generalization of the mentioned results. For given an integer vector s→=(s1,…,sk), first we define s→-stable Kneser graph KG(n,k)s→−stab as an induced subgraph of the Kneser graph KG(n,k). Then, we show that the neighborhood complex of KG(n,k)s→−stab has the same homotopy type as the n−∑i=1k−1si−2-sphere for some specific values of the parameter s→. In particular, this implies that χKG(n,k)s→−stab=n−∑i=1k−1si for those parameters. Moreover, as a simple corollary of this result, we give a lower bound on the chromatic number of 3-stable Kneser graphs which is just one less than the number conjectured in this regard.

A study of the neighborhood complex of $-stable Kneser graphs

In 1978, Alexander Schrijver defined the stable Kneser graphs as a vertex critical subgraphs of the Kneser graphs. In the early 2000s, Günter M. Ziegler generalized Schrijver’s construction and defined the s-stable Kneser graphs. Thereafter Frédéric Meunier determined the chromatic number of the s-stable Kneser graphs for special cases and formulated a conjecture on the chromatic number of the s-stable Kneser graphs. In this paper we study a generalization of the s-stable Kneser graphs. For some specific values of the parameter we show that the neighborhood complex of < s, t >-stable Kneser graph has the same homotopy type as the (t − 1)-sphere. In particular, this implies that the chromatic number of this graph is t + 1.

The super-connectivity of odd graphs and of their kronecker double cover

The study of connectivity parameters forms an integral part of the research conducted in establishing the fault tolerance of networks. A number of variations on the classical notion of connectivity have been proposed and studied. In particular, the super-connectivity asks for the minimum number of vertices that need to be deleted from a graph in order to disconnect the graph without creating isolated vertices. In this work, we determine this value for two closely related families of graphs which are considered as good models for networks, namely the odd graphs and their Kronecker double cover. The odd graphs are constructed by taking all possible subsets of size k from the set of integers {1,…,2k + 1} as vertices, and defining two vertices to be adjacent if the corresponding k-subsets are disjoint; these correspond to the Kneser graphs KG(2k + 1, k). The Kronecker double cover of a graph G is formed by taking the Kronecker product of G with the complete graph on two vertices; in the case when G is KG(2k + 1, k), the Kronecker double cover is the bipartite Kneser graph H(2k + 1, k). We show that in both instances, the super-connectivity is equal to 2k.

First-Order Complexity of Subgraph Isomorphism via Kneser Graphs

First-Order Complexity of Subgraph Isomorphism via Kneser Graphs