Schrijver Graphs Research Articles

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.

Local orthogonality dimension

AbstractAn orthogonal representation of a graph over a field is an assignment of a vector to every vertex of , such that for every vertex and whenever and are adjacent in . The locality of the orthogonal representation is the largest dimension of a subspace spanned by the vectors associated with a closed neighborhood in the graph. We introduce a novel graph parameter, called the local orthogonality dimension, defined for a given graph and a given field , as the smallest possible locality of an orthogonal representation of over . This is a local variant of the well‐studied orthogonality dimension of graphs, introduced by Lovász, analogously to the local variant of the chromatic number introduced by Erdös et al. We investigate the usefulness of topological methods for proving lower bounds on the local orthogonality dimension. Such methods are known to imply tight lower bounds on the chromatic number of several graph families of interest, such as Kneser graphs and Schrijver graphs. We prove that graphs for which topological methods imply a lower bound of on their chromatic number have local orthogonality dimension at least over every field, strengthening a result of Simonyi and Tardos on the local chromatic number. We show that for certain graphs this lower bound is tight, whereas for others, the local orthogonality dimension over the reals is equal to the chromatic number. More generally, we prove that for every complement of a line graph, the local orthogonality dimension over coincides with the chromatic number. This strengthens a recent result by Daneshpajouh, Meunier, and Mizrahi, who proved that the local and standard chromatic numbers of these graphs are equal. As another extension of their result, we prove that the local and standard chromatic numbers are equal for some additional graphs, from the family of Kneser graphs. We also study the computational aspects of the local orthogonality dimension and show that for every integer and a field , it is NP‐hard to decide whether the local orthogonality dimension of an input graph over is at most . We finally present an application of the local orthogonality dimension to the index coding problem from information theory, extending a result of Shanmugam, Dimakis, and Langberg.

The Complexity of Finding Fair Independent Sets in Cycles

Let G be a cycle graph and let \(V_1,\ldots,V_m\) be a partition of its vertex set into m sets. An independent set S of G is said to fairly represent the partition if \(|S \cap V_i| \geq \frac{1}{2} \cdot |V_i| -1\) for all \(i \in [m]\). It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al. 2017). We prove that the problem of finding such an independent set is PPA-complete. As an application, we show that the problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is PPA-complete as well. The work is motivated by the computational aspects of the `cycle plus triangles' problem and of its extensions.

On 4-chromatic Schrijver graphs: their structure, non-3-colorability, and critical edges

We investigate 4-chromatic Schrijver graphs from various points of view and color-critical edges in Schrijver graphs in general. In particular, we present the following results. Our proof of non-3-colorability of 4-chromatic Schrijver graphs is based on a complete description of their structure that we present as well. This was already also given (in somewhat different terms) by Braun and even earlier in an unpublished manuscript by Li.

On the Chromatic Number of Matching Kneser Graphs

AbstractIn an earlier paper, the present authors (2015) introduced thealtermatic numberof graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

Random Kneser Graphs and Hypergraphs

The Kneser graph $KG_{n,k}$ is the graph whose vertices are the $k$-element subsets of $[n],$ with two vertices adjacent if and only if the corresponding sets are disjoint. A famous result due to Lovász states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper we discuss the chromatic number of random Kneser graphs and hypergraphs. It was studied in two recent papers, one due to Kupavskii, who proposed the problem and studied the graph case, and the more recent one due to Alishahi and Hajiabolhassan. The authors of the latter paper had extended the result of Kupavskii to the case of general Kneser hypergraphs. Moreover, they have improved the bounds of Kupavskii in the graph case for many values of parameters.In the present paper we present a purely combinatorial approach to the problem based on blow-ups of graphs, which gives much better bounds on the chromatic number of random Kneser and Schrijver graphs and Kneser hypergraphs. This allows us to improve all known results on the topic. The most interesting improvements are obtained in the case of $r$-uniform Kneser hypergraphs with $r\ge 3$, where we managed to replace certain polynomial dependencies of the parameters by the logarithmic ones.

Independent Sets in the Union of Two Hamiltonian Cycles

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)$ is exponentially large in $n ~(n \to \infty)$. We prove that $0.26 < c_t < 0.36$, but the exact value of $c_t$ is not determined. For the lower bound we prove a technical lemma, which for graphs that are the union of two Hamiltonian cycles establishes a relation between the independence number and the number of $K_4$ subgraphs. A corollary of this lemma is that if a graph $G$ on $n>12$ vertices is the union of two Hamiltonian cycles and $\alpha(G)=n/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.

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.

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].

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.

On random subgraphs of Kneser and Schrijver graphs

A Kneser graph KGn,k is a graph whose vertices are in one-to-one correspondence with k-element subsets of [n], with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lovász states that the chromatic number of a Kneser graph KGn,k is equal to n−2k+2. In this paper we study the chromatic number of a random subgraph of a Kneser graph KGn,k as n grows. A random subgraph KGn,k(p) is obtained by including each edge of KGn,k with probability p. For a wide range of parameters k=k(n), p=p(n) we show that χ(KGn,k(p)) is very close to χ(KGn,k), w.h.p. differing by at most 4 in many cases. Moreover, we obtain the same bounds on the chromatic numbers for the so-called Schrijver graphs, which are known to be vertex-critical induced subgraphs of Kneser graphs.

Colouring quadrangulations of projective spaces

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective space Pn has chromatic number n+2 or higher, unless G is bipartite. For n=2 this was proved by Youngs (1996) [20]. The family of quadrangulations of projective spaces includes all complete graphs, all Mycielski graphs, and certain graphs homomorphic to Schrijver graphs. As a corollary, we obtain a new proof of the Lovász–Kneser theorem.

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.

Independence Complexes of Stable Kneser Graphs

For integers $n\geq 1$, $k\geq 0$, the stable Kneser graph $SG_{n,k}$ (also called the Schrijver graph) has as vertex set the stable $n$-subsets of $[2n+k]$ and as edges disjoint pairs of $n$-subsets, where a stable $n$-subset is one that does not contain any $2$-subset of the form $\{i,i+1\}$ or $\{1,2n+k\}$. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number $k+2$. This article contains a study of the independence complexes of $SG_{n,k}$ for small values of $n$ and $k$. Our contributions are two-fold: first, we prove that the homotopy type of the independence complex of $SG_{2,k}$ is a wedge of spheres of dimension two. Second, we determine the homotopy types of the independence complexes of certain graphs related to $SG_{n,2}$.

On directed local chromatic number, shift graphs, and Borsuk-like graphs

We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of “topologically t-chromatic” graphs. We show that this minimum for large enough t-chromatic Schrijver graphs and t-chromatic generalized Mycielski graphs of appropriate parameters is ⌈t/4⌉+1. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 65-82, 2010 © 2011 Wiley Periodicals, Inc.

On topological relaxations of chromatic conjectures

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad–Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in Simonyi and Tardos (2006) [41], our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number.

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 .

Non-cover generalized Mycielski, Kneser, and Schrijver graphs

A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs M m ( C 2 t + 1 ) of an odd cycle, Kneser graphs KG ( n , k ) , and Schrijver graphs SG ( n , k ) are not cover graphs when m ⩾ 0 , t ⩾ 1 , k ⩾ 1 , and n ⩾ 2 k + 2 . These results have consequences in circular chromatic number.

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.