Kneser Graphs Research Articles

Some Locally Kneser Graphs

The Kneser graph $K(n,d)$ is the graph on the $d$-subsets of an $n$-set, adjacent when disjoint. Clearly, $K(n+d,d)$ is locally $K(n,d)$. Hall showed for $n \ge 3d+1$ that there are no further examples. Here we give other examples of locally $K(n,d)$ graphs for $n = 3d$, and some further sporadic examples. It follows that Hall's bound is best possible.

A note on Pk-decomposition of the Kneser graph

A note on Pk-decomposition of the Kneser graph

Saturation in Kneser Graphs

Saturation in Kneser Graphs

Hamiltonicity of Schrijver graphs and stable Kneser graphs

For integers k≥1 and n≥2k+1, the Schrijver graph S(n,k) has as vertices all k-element subsets of [n]≔{1,2,…,n} that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers k≥1, s≥2, and n≥sk+1, the s-stable Kneser graph S(n,k,s) has as vertices all k-element subsets of [n] in which any two elements are in cyclical distance at least s. We prove that all the graphs S(n,k,s), in particular Schrijver graphs S(n,k)=S(n,k,2), admit a Hamilton cycle that can be computed in time O(n) per generated vertex.

Critical Subgraphs of Schrijver Graphs for the Fractional Chromatic Number

Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.

Gromov hyperbolicity of Johnson and Kneser graphs

The concept of Gromov hyperbolicity is a geometric concept that leads to a rich general theory. Johnson and Kneser graphs are interesting combinatorial graphs defined from systems of sets. In this work we compute the precise value of the hyperbolicity constant of every Johnson graph. Also, we obtain good bounds on the hyperbolicity constant of every Kneser graph, and in many cases, we even compute its precise value.

Failed zero forcing numbers of Kneser graphs, Johnson graphs, and hypercubes

Failed zero forcing numbers of Kneser graphs, Johnson graphs, and hypercubes

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}.

Regarding equitable colorability defect of hypergraphs

After Lovász’s break-through in determining the chromatic number of Kneser graphs (1978), and after extending this result to the chromatic number of $r$-uniform Kneser hypergraphs by Alon, Frankl, and Lovász’s (1986), some important parameters such as colorability defect and equitable colorability defect were introduced in order to provide sharp lower bounds for the chromatic number of general $r$-uniform Kneser hypergraphs. As a generalization of many earlier results in this area, Azarpendar and Jafari (2023) introduced the $s$-th equitable $r$-colorability defect ${\rm ecd}^r (\mathcal{F} , s)$; a parameter which provides a lower bound for the chromatic number of generalized Kneser hypergraphs ${\rm KG} ^r (\mathcal{F} , s)$. They proved the following nice inequality $$\chi \left( {\rm KG} ^r (\mathcal{F} , s) \right) \geq \left\lceil \frac{ {\rm ecd}^r \left( \mathcal{F} , \left\lfloor \frac{s}{2} \right\rfloor \right) }{r-1} \right\rceil ,$$ and noted that it is plausible that the above inequality remains true if one replaces $\left\lfloor \frac{s}{2} \right\rfloor$ with $s$. In this paper, considering the relation ${\rm ecd}^r \left( \mathcal{F} , x \right) \geq {\rm cd}^r \left( \mathcal{F} , x \right)$ which always holds, we show that even in the weaker inequality $$\chi \left( {\rm KG} ^r (\mathcal{F} , s) \right) \geq \left\lceil \frac{ {\rm cd}^r \left( \mathcal{F} , \left\lfloor \frac{s}{2} \right\rfloor \right) }{r-1} \right\rceil ,$$ no number $x$ greater than $\left\lfloor \frac{s}{2} \right\rfloor$ could be replaced by $\left\lfloor \frac{s}{2} \right\rfloor$.

General Position Polynomials

A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G. In this paper we introduce the general position polynomial as ∑i≥0aixi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{i \\ge 0} a_i x^i$$\\end{document}, where ai\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_i$$\\end{document} is the number of distinct general position sets of G with cardinality i. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs K(n, 2), with unimodal general position polynomials are presented.

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.

Finding the Fixing Number of Johnson Graphs J(n, k) for k Є {2; 3}

The graph invariant, aptly named the fixing number, is the smallest number of vertices that, when fixed, eliminate all non-trivial automorphisms (or symmetries) of a graph. Although many graphs have established fixing numbers, Johnson graphs, a family of graphs related to the graph isomorphism problem, have only partially classified fixing numbers. By examining specific orbit sizes of the automorphism group of Johnson graphs and classifying the subsequent remaining subgroups of the automorphism group after iteratively fixing vertices, we provide exact minimal sequences of fixed vertices, in turn establishing the fixing number of infinitely many Johnson graphs. KEYWORDS: Graph Automorphism Groups; Symmetry Breaking; Fixing Number; Determining Number; Johnson Graphs; Kneser Graphs; Graph Invariants; Permutation Groups; Minimal Sized Bases.

Proof of a conjecture on edge coloring of the Kneser graph K(t,2)

Proof of a conjecture on edge coloring of the Kneser graph K(t,2)

Neighbour-transitive codes in Kneser graphs

A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group Aut(C) acts transitively on each of the first s+1 parts C0,C1,…,Cs of the distance partition{C=C0,C1,…,Cρ}, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying set on which the Kneser graph K(n,k) is defined. Our first main result says that if C is a 2-neighbour-transitive code in K(n,k) such that C has minimum distance at least 5, then n=2k+1 (i.e., C is a code in an odd graph) and C lies in a particular infinite family or is one particular sporadic example. We then prove several results when C is a neighbour-transitive code in the Kneser graph K(n,k). First, if Aut(C) acts intransitively on Ω we characterise C in terms of certain parameters. We then assume that Aut(C) acts transitively on Ω, first proving that if C has minimum distance at least 3 then either K(n,k) is an odd graph or Aut(C) has a 2-homogeneous (and hence primitive) action on Ω. We then assume that C is a code in an odd graph and Aut(C) acts imprimitively on Ω and characterise C in terms of certain parameters. We give examples in each of these cases and pose several open problems.

Two conjectured strengthenings of Turán's theorem

We investigate two conjectured spectral graph theoretic strengthenings of Turán's theorem. Let μ1≥…≥μn denote the eigenvalues of a graph G with n vertices, m edges and clique number ω(G).The concise version of Turán's theorem is that n/(n−d) is a lower bound for the clique number ω(G), where d is the average degree. Our first conjecture is that d can be replaced in this bound with s+, where s+ is the sum of the squares of the positive eigenvalues. We prove this conjecture for triangle-free, weakly perfect and Kneser graphs and for almost all graphs. We have also used various software tools to search for a counter-example.Nikiforov proved a spectral version of Turán's theorem thatμ12≤2m(ω(G)−1)ω(G), and Bollobás and Nikiforov conjectured that for G≠Knμ12+μ22≤2m(ω(G)−1)ω(G). For our second conjecture, we propose that for all graphs (μ12+μ22) in this inequality can be replaced by the sum of the squares of the ω(G) largest eigenvalues, provided they are positive. We prove the conjecture for weakly perfect, Kneser, and classes of strongly regular graphs. We also provide experimental evidence and describe how the bound can be applied. Liu and Ning [19] published a wide-ranging paper entitled “Unsolved Problems in spectral graph theory”, and these two conjectures were placed second and fourth in their list of such problems.

On the intersection density of the Kneser graph [formula omitted

A set F⊂Sym(V) is intersecting if any two of its elements agree on some element of V. Given a finite transitive permutation group G≤Sym(V), the intersection densityρ(G) is the maximum ratio |F||V||G| where F runs through all intersecting sets of G. The intersection densityρ(X) of a vertex-transitive graph X=(V,E) is equal to maxρ(G):G≤Aut(X),G transitive. In this paper, we study the intersection density of the Kneser graph K(n,3), for n≥7. The intersection density of K(n,3) is determined whenever its automorphism group contains PSL2(q), with some exceptional cases depending on the congruence of q. We also briefly consider the intersection density of K(n,2) for values of n where PSL2(q) is a subgroup of its automorphism group.

Towards the Chen-Raspaud conjecture

For an integer k, a homomorphism from a graph G to the Kneser graph K(2k+1,k) is equivalent to assigning to each vertex of G a k-subset of {1,…,2k+1} in a way that adjacent vertices receive disjoint subsets.Chen and Raspaud (2010) [5] conjectured that for every k≥2, every graph G with maximum average degree less than 2k+1k and no odd cycles with fewer than 2k+1 vertices admits a homomorphism to K(2k+1,k). They also showed that the statement is true for k=2. In this note we confirm the conjecture for k=3.

Monochromatic spanning trees and matchings in ordered complete graphs

AbstractWe study two well‐known Ramsey‐type problems for (vertex‐)ordered complete graphs. Two independent edges in ordered graphs can be nested, crossing, or separated. Apart from two trivial cases, these relations define six types of subgraphs, depending on which one (or two) of these relations are forbidden. Our first target is to refine a remark by Erdős and Rado that every 2‐coloring of the edges of a complete graph contains a monochromatic spanning tree. We show that forbidding one relation we always have a monochromatic (nonnested, noncrossing, and nonseparated) spanning tree in a 2‐edge‐colored ordered complete graph. On the other hand, if two relations are forbidden, then it is possible that we have monochromatic (nested, separated, and crossing) subtrees of size only in a 2‐colored ordered complete graph on vertices. Some of these results relate to drawings of complete graphs. For instance, the existence of a monochromatic nonnested spanning tree in 2‐colorings of ordered complete graphs verifies a more general conjecture for twisted drawings. Our second subject is to refine the Ramsey number of matchings, that is, pairwise independent edges for ordered complete graphs. Cockayne and Lorimer proved that for given positive integers , is the smallest integer with the following property: every ‐coloring of the edges of a complete graph contains a monochromatic matching with edges. We conjecture that this result can be strengthened: ‐colored ordered complete graphs on vertices contain monochromatic nonnested and also nonseparated matchings with edges. We prove this conjecture for some special cases including the following. (i) Every ‐colored ordered complete graph on vertices contains a monochromatic nonnested matching of size two ( case). We prove it by showing that the chromatic number of the subgraph of the Kneser graph induced by the nonnested 2‐matchings in an ordered complete graph on vertices is ‐chromatic. (ii) Every 2‐colored ordered complete graph on vertices contains a monochromatic nonseparated matching of size ( case). This is the hypergraph analog of a result of Kaiser and Stehlík who proved that the Kneser graph induced by the nonseparated 2‐matchings in an ordered complete graph on vertices is ‐chromatic.For nested, separated, and crossing matchings the situation is different. The smallest ensuring a monochromatic matching of size in every ‐coloring is in the first two cases and one less in the third case.

Exploring the Fascinating World of Kneser Graphs: Characteristics, Construction, and Applications

Exploring the Fascinating World of Kneser Graphs: Characteristics, Construction, and Applications

On the Super (Edge)-Connectivity of Generalized Johnson Graphs

Let [Formula: see text], [Formula: see text] and [Formula: see text] be non-negative integers. The generalized Johnson graph [Formula: see text] is the graph whose vertices are the [Formula: see text]-subsets of the set [Formula: see text], and two vertices are adjacent if and only if they intersect with [Formula: see text] elements. Special cases of generalized Johnson graph include the Kneser graph [Formula: see text] and the Johnson graph [Formula: see text]. These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph [Formula: see text] and [Formula: see text] by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text], and further show that the Kneser graph [Formula: see text] [Formula: see text] is super vertex-connected and super cyclically edge-connected.