Repetition Number of Graphs

Every n-vertex graph has two vertices with the same degree (if n ≥ 2). In general, let rep(G) be the maximum multiplicity of a vertex degree in G. An easy counting argument yields rep(G) ≥ n/(2d − 2s + 1), where d is the average degree and s is the minimum degree of G. Equality can hold when 2d is an integer, and the bound is approximately sharp in general, even when G is restricted to be a tree, maximal outerplanar graph, planar triangulation, or claw-free graph. Among large claw-free graphs, repetition number 2 is achievable, but if G is an n-vertex line graph, then rep(G) ≥ 1 4n. Among line graphs of trees, the minimum repetition number is Θ(n1/2). For line graphs of maximal outerplanar graphs, trees with perfect matchings, or triangulations with 2-factors, the lower bound is linear.