Kneser Graphs Research Articles

The list distinguishing number of Kneser graphs

A graph G is said to be k-distinguishable if every vertex of the graph can be colored from a set of k colors such that no non-trivial automorphism fixes every color class. The distinguishing number D(G) is the least integer k for which G is k-distinguishable. If for each v∈V(G) we have a list L(v) of colors, and we stipulate that the color assigned to vertex v comes from its list L(v) then G is said to be L-distinguishable where L={L(v)}v∈V(G). The list distinguishing number of a graph, denoted Dl(G), is the minimum integer k such that every collection of lists L with |L(v)|=k admits an L-distinguishing coloring. In this paper, we prove that Dl(G)=D(G) when G is a Kneser graph.

A new approach to the chromatic number of the square of Kneser graph [formula omitted

The vertices of Kneser graph K(n,k) are the subsets of {1,2,…,n} of cardinality k, two vertices are adjacent if and only if they are disjoint. The square G2 of a graph G is defined on the vertex set of G with two vertices adjacent if their distance in G is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when n=2k+1. It is believed that χ(K2(2k+1,k))=2k+c where c is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: 8k∕3+20∕3 for 1k≥3 (Kim and Park, 2014) and 32k∕15+32 for k≥7 (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to χ(K2(2k+1,k))≤5k∕2+c, where c is a constant in {5∕2,9∕2,5,6}, depending on k≥2.

More odd graph theory from another point of view

The Kneser graph K(n,k) has as vertices all k-element subsets of [n]={1,2,…,n} and an edge between any two vertices that are disjoint. If n=2k+1, then K(n,k) is called an odd graph. Let n>4 and 1<k<n2. In the present paper, we show that if the Kneser graph K(n,k) is of even order where n is an odd integer or both of the integers n,k are even, then K(n,k) is a vertex-transitive non Cayley graph. Although, these are special cases of Godsil [7], unlike his proof that uses some very deep group-theoretical facts, ours uses no heavy group-theoretic facts. We obtain our results by using some rather elementary facts of number theory and group theory. We show that ‘almost all’ odd graphs are of even order, and consequently are vertex-transitive non Cayley graphs. Finally, we show that if k>4 is an even integer such that k is not of the form k=2t for some t>2, then the line graph of the odd graph Ok+1 is a vertex-transitive non Cayley graph.

Chromatic number of random Kneser hypergraphs

Recently, Kupavskii (2016) [20] investigated the chromatic number of random Kneser graphs KGn,k(ρ) and proved that, in many cases, the chromatic number of the random Kneser graph KGn,k(ρ) and the Kneser graph KGn,k are almost surely closed. He also marked the studying of the chromatic number of random Kneser hypergraphs KGn,kr(ρ) as a very interesting problem. With the help of Zp-Tucker lemma, a combinatorial generalization of the Borsuk–Ulam theorem, we generalize Kupavskii's result to random general Kneser hypergraphs by introducing an almost surely lower bound for the chromatic number of them. Roughly speaking, as a special case of our result, we show that the chromatic numbers of the random Kneser hypergraph KGn,kr(ρ) and the Kneser hypergraph KGn,kr are almost surely closed in many cases. Moreover, restricting to the Kneser and Schrijver graphs, we present a purely combinatorial proof for an improvement of Kupavskii's result.Also, for any hypergraph H, we present a lower bound for the minimum number of colors required in a coloring of KGr(H) with no monochromatic Kt,…,tr subhypergraph, where Kt,…,tr is the complete r-uniform r-partite hypergraph with tr vertices such that each of its parts has t vertices. This result generalizes the lower bound for the chromatic number of KGr(H) found by Alishahi and Hajiabolhassan (2015) [3].

On the chromatic number of a random subgraph of the Kneser graph

Results are obtained that substantially strengthen a previously known bound for the chromatic number of a random subgraph of the Kneser graph.

Kneser ranks of random graphs and minimum difference representations

Every graph G=(V,E) is an induced subgraph of some Kneser graph of rank k, i.e., there is an assignment of (distinct) k-sets v↦Av to the vertices v∈V such that Au and Av are disjoint if and only if uv∈E. The smallest such k is called the Kneser rank of G and denoted by fKneser(G). As an application of a result of Frieze and Reed concerning the clique cover number of random graphs we show that for constant 0<p<1 there exist constants ci=ci(p)>0, i=1,2 such that with high probabilityc1n/(log⁡n)<fKneser(G)<c2n/(log⁡n). We apply this to other graph representations defined by Boros, Gurvich and Meshulam.A k-min-difference representation of a graph G is an assignment of a set Ai to each vertex i∈V(G) such that ij∈E(G)⇔min⁡{|Ai\\Aj|,|Aj\\Ai|}≥k. The smallest k such that there exists a k-min-difference representation of G is denoted by fmin(G). Balogh and Prince proved in 2009 that for every k there is a graph G with fmin(G)≥k. We prove that there are constants c1″,c2″>0 such that c1″n/(log⁡n)<fmin(G)<c2″n/(log⁡n) holds for almost all bipartite graphs G on n+n vertices.

Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

An embedding i mapsto p_iin mathbb {R}^d of the vertices of a graph G is called universally completable if the following holds: For any other embedding imapsto q_i~in mathbb {R}^{k} satisfying q_i^{T}q_j = p_i^{T}p_j for i = j and i adjacent to j, there exists an isometry mapping q_i to p_i for all iin V(G). The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of G, which we call least eigenvalue frameworks. We identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on mathbb {Z}_2^n (n le 5) show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work we study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lovász theta number (of the complementary graph) admits a unique optimal solution. We identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and q-Kneser graphs are UVC. Lastly, we show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore, we are able to characterize all optimal vector colorings of such graphs. In particular, we give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable.

Tight Just Chromatic Excellence In Fuzzy Graphs

Let G be a simple fuzzy graph. A family  I“a¶ = { I³1, I³2,…, I³k} of fuzzy sets on a set V is called k-fuzzy colouring of V = (V,Iƒ,µ) if i) âˆa I“a¶ = Iƒ, ii) I³i∩ I³j = Ф, iii) for every strong edge (x,y) (i.e., µ(xy) > 0) of G min{I³i(x), I³j(y)} = 0; (1 ≤ i ≤ k). The minimum number of k for which there exists a k-fuzzy colouring is called the fuzzy chromatic number of G denoted as I‡f (G). Then I“a¶  is the partition of independent sets of vertices of G in which each sets has the same colour is called the fuzzy chromatic partition. A graph G is called the just I‡f -excellent if every vertex of G appears as a singleton in exactly one _f -partition of G. A just I‡f –excellent graph of order n is called the tight just I‡f -excellent if G having exactly n, I‡f -partitions. This paper aims at the study of the new concept namely tight just Chromatic excellence in fuzzy graphs and its properties. 02000 Mathematics Subject Classification:05C72 Key words: fuzzy just chromatic excellent, tight just I‡f -excellent, fuzzy colourful vertex, fuzzy kneser 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}.

On the bandwidth of the Kneser graph

Let G=(V,E) be a graph on n vertices and f:V→[1,n] a one to one map of V onto the integers 1 through n. Let dilation(f)= max{|f(v)−f(w)|:vw∈E}. Define the bandwidthB(G) of G to be the minimum possible value of dilation(f) over all such one to one maps f. Next define the Kneser GraphK(n,r) to be the graph with vertex set [n]r, the collection of r-subsets of an n element set, and edge set E={AB:A,B∈[n]r,A∩B=∅}. For fixed r≥4 and n growing we show that B(K(n,r))=n−1r+12n−4r−1−12n−1r−2+O(nr−4).

The automorphism group of the s-stable Kneser graphs

For k,s≥2, the s-stable Kneser graphs are the graphs with vertex set the k-subsets S of {1,…,n} such that the circular distance between any two elements in S is at least s and two vertices are adjacent if and only if the corresponding k-subsets are disjoint. Braun showed that for n≥2k+1 the automorphism group of the 2-stable Kneser graphs (Schrijver graphs) is isomorphic to the dihedral group of order 2n. In this paper we generalize this result by proving that for s≥2 and n≥sk+1 the automorphism group of the s-stable Kneser graphs also is isomorphic to the dihedral group of order 2n.

Identifying codes and watching systems in Kneser graphs

Identifying code in graph [Formula: see text] is a subset [Formula: see text] of [Formula: see text] such that [Formula: see text] for every [Formula: see text] of [Formula: see text] and [Formula: see text] for every [Formula: see text] of [Formula: see text]. The minimum size of identifying codes of graph [Formula: see text] is denoted by [Formula: see text]. A watching system in a graph [Formula: see text] is a set [Formula: see text], where [Formula: see text] and [Formula: see text] is a subset of closed neighborhood of [Formula: see text] such that the sets [Formula: see text] are nonempty and distinct, for any [Formula: see text]. The minimum size of a watching system of [Formula: see text] is denoted by [Formula: see text]. In this paper, we show if [Formula: see text], then [Formula: see text] if [Formula: see text] (mod 3) and [Formula: see text] if [Formula: see text] (mod 3). Also we show that [Formula: see text]. This means that in this family of graphs the watching system is more efficient than identifying code.

Formula omitted]-Labeling of Kneser graphs and coloring squares of Kneser graphs

The frequency assignment problem is to assign a frequency to each radio transmitter so that transmitters are assigned frequencies with allowed separations. Motivated by a variation of the frequency assignment problem, the L(2,1)-labeling problem was put forward. An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has a L(2,1)-labeling with max{f(v):v∈V(G)}=k. Griggs and Yeh (1992) conjectured that λ(G)≤Δ2 for any graph with maximum degree Δ≥2. The Kneser graphK(a,b) is defined as the graph whose vertices correspond to all b-subsets of the a-set A={1,2,…,a}, with edges joining pairs of vertices that correspond to non-overlapping b-subsets. The chromatic number of the Kneser graph K(a,b) is a−2b+2. Füredi put forward an open problem: What is the value for the chromatic number of the square of any Kneser graph K(a,b) (the square of a graph is the graph obtained by adding edges joining vertices at distance 2)? In this article, the L(2,1)-labeling numbers of Kneser graphs K(a,b) are considered. The combined upper bounds for the L(2,1)-labeling numbers of Kneser graphs are derived by using two approaches. The exact L(2,1)-labeling number of Kneser graph K(a,b) for a≥3b−1 is obtained, and it is proved that Griggs and Yeh’s conjecture holds for Kneser graphs and the L(2,1)-labeling numbers of Kneser graphs are much better than Δ2 in most cases. We also provide bounds for the chromatic number of the square of any Kneser graph K(a,b) using the proof for the upper bounds of the L(2,1)-labeling numbers of Kneser graphs.

Homomorphism bounds and edge-colourings of K4-minor-free graphs

We present a necessary and sufficient condition for a graph of odd-girth 2k+1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k+1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism to the projective hypercube of dimension 2k. This supports a conjecture of the third author which generalizes the four-colour theorem and relates to several outstanding conjectures such as Seymour's conjecture on edge-colourings of planar graphs. Strengthening this result, we show that the Kneser graph K(2k+1,k) satisfies the conditions, thus implying that every K4-minor free graph of odd-girth 2k+1 has fractional chromatic number exactly 2+1k. Knowing that a smallest bound of odd-girth 2k+1 must have at least (k+22) vertices, we build nearly optimal bounds of order 4k2. Furthermore, we conjecture that the suprema of the fractional and circular chromatic numbers for K4-minor-free graphs of odd-girth 2k+1 are achieved by a same bound of odd-girth 2k+1. If true, this improves, in the homomorphism order, earlier tight results on the circular chromatic number of K4-minor-free graphs. We support our conjecture by proving it for the first few cases. Finally, as an application of our work, and after noting that Seymour provided a formula for calculating the edge-chromatic number of K4-minor-free multigraphs, we show that stronger results can be obtained in the case of K4-minor-free regular multigraphs.

How long can one bluff in the domination game?

The domination game is played on an arbitrary graph G by two players, Dominator and Staller. The game is called Game 1 when Dominator starts it, and Game 2 otherwise. In this paper bluff graphs are introduced as the graphs in which every vertex is an optimal start vertex in Game 1 as well as in Game 2. It is proved that every minus graph (a graph in which Game 2 finishes faster than Game 1) is a bluff graph. A non-trivial infinite family of minus (and hence bluff) graphs is established. minus graphs with game domination number equal to 3 are characterized. Double bluff graphs are also introduced and it is proved that Kneser graphs K(n, 2), n ≥ 6, are double bluff. The domination game is also studied on generalized Petersen graphs and on Hamming graphs. Several generalized Petersen graphs that are bluff graphs but not vertex-transitive are found. It is proved that Hamming graphs are not double bluff.

Intersection patterns of finite sets and of convex sets

The main result is a common generalization of results on lower bounds for the chromatic number of r r -uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser’s conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant maps yields lower bounds for chromatic numbers. We obtain an essentially elementary proof of the result of Schrijver on the chromatic number of stable Kneser graphs. In fact, we show that every neighborly even-dimensional polytope yields a small induced subgraph of the Kneser graph of the same chromatic number. We furthermore use this geometric viewpoint to give tight lower bounds for the chromatic number of certain small subhypergraphs of Kneser hypergraphs.

Shifts of the stable Kneser graphs and hom-idempotence

A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n≥1 there is a homomorphism from Gn+1 to Gn. Larose et al. (1998) proved that Kneser graphs KG(n,k) are not weakly hom-idempotent for n≥2k+1, k≥2. For s≥2, we characterize all the shifts (i.e., automorphisms of the graph that map every vertex to one of its neighbors) of s-stable Kneser graphs KG(n,k)s−stab and we show that 2-stable Kneser graphs are not weakly hom-idempotent, for n≥2k+2, k≥2. Moreover, for s,k≥2, we prove that s-stable Kneser graphs KG(ks+1,k)s−stab are circulant graphs and so hom-idempotent graphs. Finally, for s≥3, we show that s-stable Kneser graphs KG(2s+2,2)s−stab are cores, not χ-critical, not hom-idempotent and their chromatic number is equal to s+2.

On the stability of some Erdős–Ko–Rado type results

Consider classical Kneser’s graph K(n,r): for two natural numbers r,n such that r⩽n∕2, its vertices are all the subsets of [n]={1,2,…,n} of size r, and two such vertices are adjacent if the corresponding subsets are disjoint. The Erdős–Ko–Rado theorem states that the size of the largest independent set in this graph is n−1r−1. Now let us delete each edge of the graph K(n,r) with some fixed probability p independently of each other. Quite surprisingly, the independence number of such random subgraph Kp(n,r) of the graph K(n,r) is, with high probability, the same as the independence number of the initial graph. This phenomenon is called the stability of the independence number. This paper concerns the independence number of random subgraphs of the graph G(n,r,1), which vertices are the same as in the Kneser graph and there is an edge between two vertices if they intersect in exactly one element. We prove the stability result for constant r and p=12. To do that, we show that any sufficiently large independent set in G(n,r,1) has some predefined structure, and this is a result of independent interest.

Bipartite Kneser Graphs are Hamiltonian

For integers $k\geq 1$ and $n\geq 2k+1$ the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as vertices all $k$-element and $(n-k)$-element subsets of $[n]$ and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph $K(5,2)$ have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs $H(n,k)$. We also establish the existence of cycles that visit almost all vertices in Kneser graphs $K(n,k)$ when $n=2k+o(k)$, generalizing and improving upon previous results on this problem.

The achromatic number of Kneser graphs

The achromatic number α of a graph is the largest number of colors that can be assigned to its vertices such that adjacent vertices have different color and every pair of different colors appears on the end vertices of some edge.We estimate the achromatic number of Kneser graphs K(n,k) and determine α(K(n,k)) for some values of n and k. Furthermore, we study the achromatic number of some geometric type Kneser graphs.