The determining number of Kneser graphs

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

A formula for computing the exact determining number of Kneser graphs

openA 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. This means that, for any two automorphisms s a and b of G, if for each s in S, we have a(s)=b(s), then a=b. The determining number of G is the minimum cardinality of a determining set of G. We study the determining number of Kneser graphs K_{n,k}. In this case, the determining number equals the base size of the symmetric group S_n of degree n in its action on the k-subsets of {1,...,n}, that is, the minimum number of k-subsets such that the only permutation of {1,...,n} that fixes them all is the identity. We prove a formula that allows us to compute the determining number, and hence the base size of this action, for every n and k.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. This means that, for any two automorphisms s a and b of G, if for each s in S, we have a(s)=b(s), then a=b. The determining number of G is the minimum cardinality of a determining set of G. We study the determining number of Kneser graphs K_{n,k}. In this case, the determining number equals the base size of the symmetric group S_n of degree n in its action on the k-subsets of {1,...,n}, that is, the minimum number of k-subsets such that the only permutation of {1,...,n} that fixes them all is the identity. We prove a formula that allows us to compute the determining number, and hence the base size of this action, for every n and k

Colouring Lines in Projective Space

Let $V$ be a vector space of dimension $v$ over a field of order $q$. The $q$-Kneser graph has the $k$-dimensional subspaces of $V$ as its vertices, where two subspaces $\alpha$ and $\beta$ are adjacent if and only if $\alpha\cap\beta$ is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers of these graphs. This problem is trivial when $k=1$ (and the graphs are complete) or when $v<2k$ (and the graphs are empty). We establish some basic theory in the general case. Then specializing to the case $k=2$, we show that the chromatic number is $q^2+q$ when $v=4$ and $(q^{v-1}-1)/(q-1)$ when $v > 4$. In both cases we characterise the minimal colourings.Comment: 19 pages; to appear in J. Combinatorial Theory, Series

The Determining Number and Cost of 2-Distinguishing of Select Kneser Graphs

A graph $G$ is said to be \emph{d-distinguishable} if there exists a not-necessarily proper coloring with $d$ colors such that only the trivial automorphism preserves the color classes. For a 2-distinguishing labeling, the \emph{ cost of $2$-distinguishing}, denoted $\rho(G),$ is defined as the minimum size of a color class over all $2$-distinguishing colorings of $G$. Our work also utilizes \emph{determining sets} of $G,$ sets of vertices $S \subseteq G$ such that every automorphism of $G$ is uniquely determined by its action on $S.$ The \emph{determining number} of a graph is the size of a smallest determining set. We investigate the cost of $2$-distinguishing families of Kneser graphs $K_{n:k}$ by using optimal determining sets of those families. We show the determining number of \kntwo is equal to $\left\lceil{ \frac{2n-2}{3}}\right\rceil$and give linear bounds on \rho(\kntwo) when $n$ is sufficiently sized

Resolving sets for Johnson and Kneser graphs

A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. 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.Comment: 23 pages, 2 figures, 1 tabl