Induced subgraphs of Johnson graphs

Abstract

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, includ-ing: 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.