Kneser Graphs Research Articles

Paths of homomorphisms from stable Kneser graphs

We denote by SG n,k the stable Kneser graph (Schrijver graph) of stable n-subsets of a set of cardinality 2n+k. For k?3 (mod 4) and n?2 we show that there is a component of the ?-colouring graph of SG n,k which is invariant under the action of the automorphism group of SG n,k . We derive that there is a graph G with ?(G)=?(SG n,k ) such that the complex Hom(SG n,k ,G) is non-empty and connected. In particular, for k?3 (mod 4) and n?2 the graph SG n,k is not a test graph.

Maximal cocliques in the Kneser graph on point–plane flags in [formula omitted

We determine the maximal cocliques of size ≥5q2+5q+2 in the Kneser graph on point–plane flags in PG(4,q). The maximal size of a coclique in this graph is (q2+q+1)(q3+q2+q+1).

On the Shannon Capacity of Triangular Graphs

The Shannon capacity of a graph $G$ is $c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of $G$. The Shannon capacity of the Kneser graph $\mathrm{KG}_{n,r}$ was determined by Lovász in 1979, but little is known about the Shannon capacity of the complement of that graph when $r$ does not divide $n$. The complement of the Kneser graph, $\overline{\mathrm{KG}}_{n,2}$, is also called the triangular graph $T_n$. The graph $T_n$ has the $n$-cycle $C_n$ as an induced subgraph, whereby $c(T_n) \geq c(C_n)$, and these two families of graphs are closely related in the current context as both can be considered via geometric packings of the discrete $d$-dimensional torus of width $n$ using two types of $d$-dimensional cubes of width $2$. Bounds on $c(T_n)$ obtained in this work include $c(T_7) \geq \sqrt[3]{35} \approx 3.271$, $c(T_{13}) \geq \sqrt[3]{248} \approx 6.283$, $c(T_{15}) \geq \sqrt[4]{2802} \approx 7.276$, and $c(T_{21}) \geq \sqrt[4]{11441} \approx 10.342$.

Counting Families of Mutually Intersecting Sets

We show that the number of maximal intersecting families on a 9-set equals 423295099074735261880, that the number of independent sets of the Kneser graph K(9,4) equals 366996244568643864340, and that the number of intersecting families on an 8-set and on a 9-set is 14704022144627161780744368338695925293142507520 and 12553242487940503914363982718112298267975272720808010757809032705650591023015520462677475328 (roughly 1.255 . 10^91), respectively.

The Distinguishing Chromatic Number of Kneser Graphs

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 $r\ge 3$.

The determining number of Kneser graphs

Graph Theory 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.

Resolving sets for Johnson and Kneser graphs

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)≠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 to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.

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.

A special case of the Stahl conjecture

Let Gn,k denote the Kneser graph whose vertices are the n-element subsets of a (2n+k)-element set and whose edges are the disjoint pairs. In this paper we prove that for any non-negative integer s there is no graph homomorphism from G4,2 to G4s+1,2s+1. This confirms a conjecture of Stahl in a special case.

A short proof for Chenʼs Alternative Kneser Coloring Lemma

We give a short proof for Chenʼs Alternative Kneser Coloring Lemma. This leads to a short proof for the Johnson–Holroyd–Stahl conjecture that Kneser graphs have their circular chromatic numbers equal to their chromatic numbers.

On real number labelings and graph invertibility

For non-negative real x0 and simple graph G, λx0,1(G) is the minimum span over all labelings that assign real numbers to the vertices of G such that adjacent vertices receive labels that differ by at least x0 and vertices at distance two receive labels that differ by at least 1. In this paper, we introduce the concept of λ-invertibility: G is λ-invertible if and only if for all positive x, λx,1(G)=xλ1x,1(Gc). We explore the conditions under which a graph is λ-invertible, and apply the results to the calculation of the function λx,1(G) for certain λ-invertible graphs G. We give families of λ-invertible graphs, including certain Kneser graphs, line graphs of complete multipartite graphs, and self-complementary graphs. We also derive the complete list of all λ-invertible graphs with maximum degree 3.

Arrangements of [formula omitted]-sets with intersection constraints

Given positive integers n,k where n≥k, let f(n,k) denote the largest integer s such that there exists a cyclic ordering of the k-sets on [n]={0,1,…,n−1} such that every s consecutive k-sets are pairwise intersecting. Equivalently, f(n,k) is the largest s such that the complement K(n,k)¯ of the Kneser graph K(n,k) contains the sth power of a Hamiltonian cycle.For each n≥6 we show that f(n,2)=3. We show that f(n,3) equals either 2n−8 or 2n−7 when n is sufficiently large, conjecturing that 2n−8 is the correct value. For each k≥4 and n sufficiently large we show that 2nk−2(k−2)!−(72k−2)nk−3(k−3)!−O(nk−4)≤f(n,k)≤2nk−2(k−2)!−(72k−3.2)nk−3(k−3)!+o(nk−3).

Erdös-Ko-Rado from intersecting shadows

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdýos–Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corresponding results.

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.

The eigenvalues of [formula omitted]-Kneser graphs

In this note, by proving some combinatorial identities, we obtain a simple form for the eigenvalues of q -Kneser graphs.

Formula omitted]-coloring of Kneser graphs

A b -coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b -coloring using k colors is the b -chromatic number b ( G ) of G . The b -spectrum S b ( G ) of a graph G is the set of positive integers k , χ ( G ) ≤ k ≤ b ( G ) , for which G has a b -coloring using k colors. A graph G is b -continuous if S b ( G ) = the closed interval [ χ ( G ) , b ( G ) ] . In this paper, we obtain an upper bound for the b -chromatic number of some families of Kneser graphs. In addition we establish that [ χ ( G ) , n + k + 1 ] ⊂ S b ( G ) for the Kneser graph G = K ( 2 n + k , n ) whenever 3 ≤ n ≤ k + 1 . We also establish the b -continuity of some families of regular graphs which include the family of odd graphs.

Häggkvist–Hell graphs: A class of Kneser-colorable graphs

For positive integers n and r we define the Häggkvist–Hell graph, H n : r , to be the graph whose vertices are the ordered pairs ( h , T ) where T is an r -subset of [ n ] , and h is an element of [ n ] not in T . Vertices ( h x , T x ) and ( h y , T y ) are adjacent iff h x ∈ T y , h y ∈ T x , and T x ∩ T y = ∅ . These triangle-free arc transitive graphs are an extension of the idea of Kneser graphs, and there is a natural homomorphism from the Häggkvist–Hell graph, H n : r , to the corresponding Kneser graph, K n : r . Häggkvist and Hell introduced the r = 3 case of these graphs, showing that a cubic graph admits a homomorphism to H 22 : 3 if and only if it is triangle-free. Gallucio, Hell, and Nes˘etr˘il also considered the r = 3 case, proving that H n : 3 can have arbitrarily large chromatic number. In this paper we give the exact values for diameter, girth, and odd girth of all Häggkvist–Hell graphs, and we give bounds for independence, chromatic, and fractional chromatic number. Furthermore, we extend the result of Gallucio et al. to any fixed r ≥ 2 , and we determine the full automorphism group of H n : r , which is isomorphic to the symmetric group on n elements.

An Inductive Construction for Hamilton Cycles in Kneser Graphs

The Kneser graph $K(n,r)$ has as vertices all $r$-subsets of an $n$-set with two vertices adjacent if the corresponding subsets are disjoint. It is conjectured that, except for $K(5,2)$, these graphs are Hamiltonian for all $n\geq 2r+1$. In this note we describe an inductive construction which relates Hamiltonicity of $K(2r+2s,r)$ to Hamiltonicity of $K(2r'+s,r')$. This shows (among other things) that Hamiltonicity of $K(2r+1,r)$ for all $3\leq r\leq k$ implies Hamiltonicity of $K(2r+2,r)$ for all $r\leq 2k+1$. Applying this result extends the range of values for which Hamiltonicity of $K(n,r)$ is known. Another consequence is that certain families of Kneser graphs ($K(\frac{27}{13}r,r)$ for instance) contain infinitely many Hamiltonian graphs.

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 .

On the locating chromatic number of Kneser graphs

Let c be a proper k -coloring of a connected graph G and Π = ( C 1 , C 2 , … , C k ) be an ordered partition of V ( G ) into the resulting color classes. For a vertex v of G , the color code of v with respect to Π is defined to be the ordered k -tuple c Π ( v ) : = ( d ( v , C 1 ) , d ( v , C 2 ) , … , d ( v , C k ) ) , where d ( v , C i ) = min { d ( v , x ) | x ∈ C i } , 1 ≤ i ≤ k . If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G , denoted by χ L ( G ) . In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ L ( K G ( n , 2 ) ) = n − 1 for all n ≥ 5 . Then, we prove that χ L ( K G ( n , k ) ) ≤ n − 1 , when n ≥ k 2 . Moreover, we present some bounds for the locating chromatic number of odd graphs.