Generalised outerplanar Turán numbers and maximum number of k-vertex subtrees

Turán numbers of vertex-disjoint cliques in -partite graphs

On two Turán Numbers

Let Hð3;1Þ denote the complete bipartite graph K3;3 with an edge deleted. For given graphs G and F , the Turán numbers exðG; FÞ denote the maximum number of edges in a subgraph of G containing no copy of F . We prove that exðKn;n;Hð3; 1ÞÞ ð4= ffiffiffi 7 p n þOðnÞ 1:5119n þOðnÞ and that exðKn;Hð3; 1ÞÞ ð1=5Þ ffiffiffiffiffi 15 p n þOðnÞ 0:7746n þ OðnÞ, which improve earlier results of Mubayi-Wes...

Vertex Degrees in Outerplanar Graphs

For an outerplanar graph on n vertices, we determine the maximum number of vertices of degree at least k. For k = 4 (and n ≥ 7) the answer is n−4. For k = 5 (and n ≥ 4), the answer is ⌊

Mean Ramsey-Turán numbers

A ρ-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most ρ. For a graph H and for ρ ≥ 1, the mean RamseyTurán number RT (n,H, ρ − mean) is the maximum number of edges a ρ-mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H . It is conjectured that RT (n,Km, 2 −mean) ...

Induced Turán Numbers

The Kővari-Sós-Turán theorem says that if G is an n-vertex graph with no Ks,t as a subgraph, then e(G) = O(n 2−1/s). Our main theorem states that if H is any fixed graph, and if G is H-free and does not contain Ks,t as an induced subgraph, then still e(G) = O(n2−1/s). A key step in our proof is to give a nontrivial upper bound on the number of cliques of fixed size in a Kr-free graph with no in...