Abstract Given an $$(r + 1)$$ ( r + 1 ) -chromatic graph H , the fundamental edge stability result of Erdős and Simonovits says that all n -vertex -free graphs have at most $$(1 - 1/r o(1)) \binom{n}{2}$$ -</mml...

Let Dk be the class of graphs for which every minor has minimum degree at most k. Then Dk is closed under taking minors. By the Robertson-Seymour graph minor theorem, Dk is characterised by a finite family of minor-minimal forbidden graphs, which we denote by D̂k. This paper discusses D̂k and related topics. We obtain four main results: 1. We prove that every (k + 1)-regular graph with less than ...

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. Krivelevich and Yuster have shown that a connected graph G with n vertices has rc(G) < 20n δ(G) [M. Krivelevich and...

Given δ and n, a minimum degree game starts with n disconnected nodes. Two players alternate, each adding a new edge in turn, until the resulting graph has minimum degree at least δ. In the achievement game, the last player to move is the winner; in the avoidance game, the last to play is the loser. We determine a winning strategy for the avoidance game for every δ and n. The achievement game i...