500 research outputs found
Induced Subgraphs of Johnson Graphs
The Johnson graph J(n,N) is defined as the graph whose vertices are the
n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share
exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson
graphs (JIS for short) do not seem to have been studied before. We give some
necessary conditions and some sufficient conditions for a graph to be JIS,
including: in a JIS graph, any two maximal cliques share at most two vertices;
all trees, cycles, and complete graphs are JIS; disjoint unions and Cartesian
products of JIS graphs are JIS; every JIS graph of order n is an induced
subgraph of J(m,2n) for some m <= n. This last result gives an algorithm for
deciding if a graph is JIS. We also show that all JIS graphs are edge move
distance graphs, but not vice versa.Comment: 12 pages, 4 figure
Primitive decompositions of Johnson graphs
A transitive decomposition of a graph is a partition of the edge set together
with a group of automorphisms which transitively permutes the parts. In this
paper we determine all transitive decompositions of the Johnson graphs such
that the group preserving the partition is arc-transitive and acts primitively
on the parts.Comment: 35 page
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
Isometric embeddings of Johnson graphs in Grassmann graphs
Let be an -dimensional vector space () and let
be the Grassmannian formed by all -dimensional
subspaces of . The corresponding Grassmann graph will be denoted by
. We describe all isometric embeddings of Johnson graphs
, in , (Theorem 4). As a
consequence, we get the following: the image of every isometric embedding of
in is an apartment of if and
only if . Our second result (Theorem 5) is a classification of rigid
isometric embeddings of Johnson graphs in , .Comment: New version -- 14 pages accepted to Journal of Algebraic
Combinatoric
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