On maximal paths and circuits of graphs

TL;DR: The maximum number of edges in a properly edge-colored graph on n vertices is the {\emph rainbow Tur\'an number} of F, and bounds are given on this maximum, disproving a conjecture in Keevash et al.

TL;DR: This paper almost proves that every graph of order n and size at least ?n2/4??n+59 is weakly pancyclic or bipartite.

TL;DR: The asymptotic value of the Ramsey number is found for a triple of long cycles, where the lengths of the cycles are large but may have different parity.

TL;DR: It is proved that for every integer k≥4, if n is even, then Rk(Cn)≥(k−1)n−2k+ 4, which is the smallest integer N for which for any edge-coloring of the complete graph KN by k colors there exists a color i for which the corresponding color class contains L as a subgraph.

TL;DR: The first result in this direction was due to Turân as discussed by the authors, who proved that a graph with kn vertices and Ck, 2n+1 edges always contains a complete graph of order k + 1.