Kneser Graph KG Research Articles

On the diameter of Schrijver graphs

For k≥1 and n≥2k, the well known Kneser graph KG(n,k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph SG(n,k) of KG(n,k) with the same chromatic number. In this paper, we compute the diameter of the graph SG(2k+r,k) with r≥1. We obtain an exact value of the diameter of SG(2k+r,k) when r∈{1,2} or when r≥k−3. For the remaining cases, when 3≤r≤k−4, we show that the diameter of SG(2k+r,k) belongs to the set {4,…,k−r−1}.

Dominator coloring of Kneser graph

A dominator coloring (briefly DC) of a graph G is a proper coloring such that each vertex dominates every vertex of at least one color class (possibly its own class). The dominator chromatic number χd(G) of G is the smallest cardinality of color classes in a DC of G. By using Steiner triple systems, in this paper, we establish that the dominator coloring of the Kneser graph KG(n, 2) is n for n ≥ 5.

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.

On the diameter of Schrijver graphs

For k ≥ 1 and n ≥ 2k, the well known Kneser graph KG(n, k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph SG(n, k) of KG(n, k) with the same chromatic number. In this paper, we compute the diameter of the graph SG(2k + r,k) with r ≥ 1. We obtain that the diameter of SG(2k + r, k) is equal to 2 if r ≥ 2k - 2; 3 if k≥ - 2 ≤ r ≤ 2k - 3; k if r = 1; and for 2 ≤ r ≤ k - 3, we obtain that the diameter of SG(2k + r, k) is at most equal to k - r + 1.

Edge-critical subgraphs of Schrijver graphs

For k≥1 and n≥2k, the Kneser graph KG(n,k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of KG(n,k) is n−2k+2. Schrijver constructed a vertex-critical subgraph SG(n,k) of KG(n,k) with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges, however, no analogous construction is known except in trivial cases. We provide such a construction for k=2 and arbitrary n≥4 by means of a simple explicit combinatorial definition.

On chromatic number and minimum cut

For a graph G, the tree Kneser graph KG(G,Tt) has all tree subgraphs of G with t vertices as vertex set and two vertices are adjacent if their corresponding trees are edge-disjoint. Also, the r-th cut number of G is the minimum number of edges between the parts of a partition of the vertex set of G into two parts such that each part has size at least r. In this paper, we investigate the chromatic number of tree Kneser graphs. Roughly speaking, we prove that for any nonnegative integer function t=t(n), where n−t=o(n), if n is sufficiently large, then for any dense graph G with n vertices, the chromatic number of the tree Kneser graph KG(G,Tt) is equal to the (n−t+1)-th cut number of G. In particular, as a consequence of this result, if n is large enough, then for any dense graph G with n vertices, the chromatic number of the tree Kneser graph KG(G,Tn) is equal to the minimum cut (minimum degree) of G. This result can be somehow considered a reminiscent of the Nash-Williams disjoint trees theorem. The proof is based on a lower bound for the chromatic number of graphs found by the present authors Alishahi and Hajiabolhassan (2015) [1] which is inspired by Tucker's lemma, an equivalent combinatorial version of the Borsuk-Ulam theorem. Finally, motivated by the aforementioned results, we close the paper by a discussion on the complexity of determining the r-th cut number of graphs.

Circular chromatic number of induced subgraphs of Kneser graphs

Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Hajiabolhassan and Zhu in 2003 proved that if n is sufficiently large with respect to k , then the Schrijver graph SG( n , k ) has the same chromatic and circular chromatic number. Later, Meunier in 2005 and independently, Simonyi and Tardos in 2006 proved that χ (SG( n , k )) = χ c (SG( n , k )) if n is even. In this paper, we study the circular chromatic number of induced subgraphs of Kneser graphs. In this regard, we shall first generalize the preceding result to s -stable Kneser graphs for large even n and even s . Furthermore, as a generalization of the Hajiabolhassan-Zhu result, we prove that if n is large enough with respect to k , then any sufficiently large induced subgraph of the Kneser graph KG( n , k ) has the same chromatic number and circular chromatic number.

The critical group of the Kneser graph on 2-subsets of an n-element set

In this paper we compute the critical group of the Kneser graph KG(n,2). This is equivalent to computing the Smith normal form of a Laplacian matrix of this graph.

Chromatic number via Turán number

For a graph G and a family of graphs F, the general Kneser graph KG(G,F) is a graph with the vertex set consisting of all subgraphs of G isomorphic to some member of F and two vertices are adjacent if their corresponding subgraphs are edge disjoint. In this paper, we introduce some generalizations of Turán number of graphs. In view of these generalizations, we give some lower and upper bounds for the chromatic number of general Kneser graphs KG(G,F). Using these bounds, we determine the chromatic number of some family of general Kneser graphs KG(G,F) in terms of generalized Turán number of graphs. In particular, we determine the chromatic number of every Kneser multigraph KG(G,F) where G is a multigraph each of whose edges has the multiplicity at least 2 and F is an arbitrary family of simple graphs. Moreover, the chromatic number of general Kneser graph KG(G,F) is exactly determined where G is a dense graph and F={K1,2}.

Bounds on the domination number of Kneser graphs

The Kneser graph KG n , k has one vertex for each k -subset of an n -set and edges between vertices whenever the corresponding subsets are disjoint. A dominating set in a graph G = ( V , E ) is a subset S ⊆ V such that each vertex in V \ S is adjacent to at least one vertex in S . The domination number of , denoted by γ ( n , k ) , is the minimum size of a dominating set in that graph. Combinatorial and computer-aided techniques for obtaining bounds on γ ( n , k ) are here considered, and several new bounds are obtained. An updated table of bounds on γ ( n , k ) is presented for n ≤ 21 and k ≤ 5 .

Secure frameproof codes through biclique covers

Graph Theory For a binary code Γ of length v, a v-word w produces by a set of codewords {w1,...,wr}⊆Γ if for all i=1,...,v, we have wi∈{w1i,...,wri} . We call a code r-secure frameproof of size t if |Γ|=t and for any v-word that is produced by two sets C1 and C2 of size at most r then the intersection of these sets is nonempty. A d-biclique cover of size v of a graph G is a collection of v-complete bipartite subgraphs of G such that each edge of G belongs to at least d of these complete bipartite subgraphs. In this paper, we show that for t≥2r, an r-secure frameproof code of size t and length v exists if and only if there exists a 1-biclique cover of size v for the Kneser graph KG(t,r) whose vertices are all r-subsets of a t-element set and two r-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of d-biclique covers of Kneser graphs and cover-free families, where an (r,w;d) cover-free family is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. Also, we present an upper bound for 1-biclique covering number of Kneser graphs.

Topological indices of the Kneser graph KG n,k

In this paper we use transitivity property of the automorphism group of the Kneser graph to calculate its Wiener, Szeged and PI indices.

A generalization of Kneser’s conjecture

We investigate some coloring properties of Kneser graphs. A semi-matching coloring is a proper coloring c : V ( G ) → N such that for any two consecutive colors, the edges joining the colors form a matching. The minimum positive integer t for which there exists a semi-matching coloring c : V ( G ) → { 1 , 2 , … , t } is called the semi-matching chromatic number of G and denoted by χ m ( G ) . In view of Tucker–Ky Fan’s lemma, we show that χ m ( KG ( n , k ) ) = 2 χ ( KG ( n , k ) ) − 2 = 2 n − 4 k + 2 provided that n ≤ 8 3 k . This gives a partial answer to a conjecture of Omoomi and Pourmiri [Local coloring of Kneser graphs, Discrete Mathematics, 308 (24): 5922–5927, (2008)]. Moreover, for any Kneser graph KG ( n , k ) , we show that χ m ( KG ( n , k ) ) ≥ max { 2 χ ( KG ( n , k ) ) − 10 , χ ( KG ( n , k ) ) } , where n ≥ 2 k ≥ 4 . Also, for n ≥ 2 k ≥ 4 , we conjecture that χ m ( KG ( n , k ) ) = 2 χ ( KG ( n , k ) ) − 2 .

Circular coloring and Mycielski construction

In this paper, we investigate the circular chromatic number of the iterated Mycielskian of graphs. It was shown by Simonyi and Tardos [G. Simonyi, G. Tardos, Local chromatic number, Ky Fan’s theorem and circular colorings, Combinatorica 26 (5) (2006) 587–626] that the t th iterate of the Mycielskian of the Kneser graph KG ( m , n ) has the same circular chromatic number and chromatic number provided that m + t is an even integer. We prove that if m is large enough, then χ ( M t ( KG ( m , n ) ) ) = χ c ( M t ( KG ( m , n ) ) ) where M t is the t th iterate of the Mycielskian operator. Also, we consider the generalized Kneser graph KG ( m , n , s ) and show that there exists a threshold m ( n , s , t ) such that χ ( M t ( KG ( m , n , s ) ) ) = χ c ( M t ( KG ( m , n , s ) ) ) for m ≥ m ( n , s , t ) .

Symmetries of the stable Kneser graphs

It is well known that the automorphism group of the Kneser graph KG n , k is the symmetric group on n letters. For n ⩾ 2 k + 1 , k ⩾ 2 , we prove that the automorphism group of the stable Kneser graph SG n , k is the dihedral group of order 2 n.

Coloring the square of the Kneser graph [formula omitted] and the Schrijver graph [formula omitted

The Kneser graph KG ( n , k ) is the graph whose vertex set consists of all k -subsets of an n -set, and two vertices are adjacent if and only if they are disjoint. The Schrijver graph SG ( n , k ) is the subgraph of KG ( n , k ) induced by all vertices that are 2-stable subsets. The square G 2 of a graph G is defined on the vertex set of G such that distinct vertices within distance two in G are joined by an edge. The span λ ( G ) of G is the smallest integer m such that an L ( 2 , 1 ) -labeling of G can be constructed using labels belonging to the set { 0 , 1 , … , m } . The following results are established. (1) χ ( KG 2 ( 2 k + 1 , k ) ) ⩽ 3 k + 2 for k ⩾ 3 and χ ( KG 2 ( 9 , 4 ) ) ⩽ 12 ; (2) χ ( SG 2 ( 2 k + 2 , k ) ) = λ ( SG ( 2 k + 2 , k ) ) = 2 k + 2 for k ⩾ 4 , χ ( SG 2 ( 8 , 3 ) ) = 8 , λ ( SG ( 8 , 3 ) ) = 9 , χ ( SG 2 ( 6 , 2 ) ) = 9 , and λ ( SG ( 6 , 2 ) ) = 8 .

Circular chromatic number of Kneser graphs

This paper proves that for any positive integer n, if m is large enough, then the reduced Kneser graph KG 2( m, n) has its circular chromatic number equal its chromatic number. This answers a question of Lih and Liu (J. Graph Theory 41 (2002) 62). For Kneser graphs, we prove that if m⩾2 n 2( n−1), then KG( m, n) has its circular chromatic number equal its chromatic number. This provides strong support for a conjecture of Johnson, Holroyd, and Stahl (J. Graph Theory 26 (1997) 137).