A Glazman–Povzner–Wienholtz theorem on graphs

A Theorem on Planar Graphs

Ramsey’s Theorem on Graphs

Imagine that you have 6 people at a party. We assume that, for every pair of them, either THEY KNOW EACH OTHER or NEITHER OF THEM KNOWS THE OTHER. So we are assuming that if x knows y, then y knows x. Claim: Either there are at least 3 people all of whom know one another, or there are at least 3 people no two of whom know each other (or both). Proof of Claim: Let the people be p1, p2, p3, p4, p...

On a min-max theorem on bipartite graphs

Frank, Sebő and Tardos [4] proved that for any connected bipartite graft (G, T ), the minimum size of a T-join is equal to the maximum value of a partition of A, where A is one of the two colour classes of G. Their proof consists of constructing a partition of A of value |F |, by using a minimum T-join F. That proof depends heavily on the properties of distances in graphs with conservative weig...

Chasing robbers on random graphs: Zigzag theorem

In this paper, we study the vertex pursuit game of Cops and Robbers where cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n, p) for a wide range of p = p(n). It has been shown that the cop number as a function o...

On Turáns theorem for sparse graphs

a?nl(t+1) where a denotes the maximum size of an independent set in G. We improve this bound for graphs containing no large cliques. 0. Notation n=n(G)=number of vertices of the graph G e=e(G)=number of edges of G h=h(G)=number of triangles in G deg (P)=valency (degree) of the vertex P deg, (P) = triangle-valency of P=number of triangles in G adjacent to P t=t(G)=n f deg (P) = 2eln = average va...