784 research outputs found
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 be a vector space of dimension over a field of order . The
-Kneser graph has the -dimensional subspaces of as its vertices,
where two subspaces and are adjacent if and only if
is the zero subspace. This paper is motivated by the problem
of determining the chromatic numbers of these graphs. This problem is trivial
when (and the graphs are complete) or when (and the graphs are
empty). We establish some basic theory in the general case. Then specializing
to the case , we show that the chromatic number is when and
when . 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 is said to be \emph{d-distinguishable} if there exists a not-necessarily proper coloring with colors such that only the trivial automorphism preserves the color classes. For a 2-distinguishing labeling, the \emph{ cost of -distinguishing}, denoted is defined as the minimum size of a color class over all -distinguishing colorings of . Our work also utilizes \emph{determining sets} of sets of vertices such that every automorphism of is uniquely determined by its action on The \emph{determining number} of a graph is the size of a smallest determining set. We investigate the cost of -distinguishing families of Kneser graphs by using optimal determining sets of those families. We show the determining number of \kntwo is equal to and give linear bounds on \rho(\kntwo) when is sufficiently sized
Resolving sets for Johnson and Kneser graphs
A set of vertices in a graph is a {\em resolving set} for if, for
any two vertices , there exists such that the distances . In this paper, we consider the Johnson graphs and Kneser
graphs , 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
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