We almost completely solve a number of problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number k, the smallest number m = m(k) such that every digraph can be coloured with m colours where each vertex has the same colour as at most a 1/k proportion of its out-neighbours. We show that m(k) ∈ {2k − 1,2k}. We also prove a result supporting the conjecture that m(2) = 3. Moreover, we prove similar results for a more general concept called majority choosability.

In this paper, we prove that given a 2-edge-coloured complete graph K4n that has the same number of edges of each colour, we can always find a perfect matching with an equal number of edges of each colour. This solves a problem posed by Caro, Hansberg, Lauri, and Zarb. The problem is also independently solved by Ehard, Mohr, and Rautenbach.

Abstract. We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every n-vertex graph admits a separating path system of size O(n) and prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs with linear minimum degree. We also obtain tight bounds on the size of a minimal separating path system in the case of trees.

We prove that if an n-vertex graph with minimum degree at least 3 contains a Hamiltonian cycle, then it contains another cycle of length n − o(n); this implies, in particular, that a well-known conjecture of Sheehan from 1975 holds asymptotically. Our methods, which combine constructive, poset-based techniques and non-constructive, parity-based arguments, may be of independent interest.

a b s t r a c tA triple of vertices in a graph is a frustrated triangle if it induces an odd number of edges.We study the set F n ⊂ [0,  n 3  ] of possible number of frustrated triangles f (G) in a graph G on n vertices. We prove that about two thirds of the numbers in [0, n 3/2 ] cannot appear in F n , and we characterise the graphs G with f (G) ∈ [0, n 3/2]. More precisely, our main result is that, for each n ≥ 3, F n contains two interlacing sequences 0and only if G can be obtained from a complete bipartite graph by flipping exactly t edges/nonedges. On the other hand, we show, for all n sufficiently large, that if m, then m ∈ F n where f (n) is asymptotically best possible with f (n) ∼ n 3/2 for n even and f (n) ∼ √ 2n 3/2 for n odd. Furthermore, we determine the graphs with the minimum number of frustrated triangles amongst those with n vertices and e ≤ n 2 /4 edges.