14,428 research outputs found
On path-quasar Ramsey numbers
Let and be two given graphs. The Ramsey number is
the least integer such that for every graph on vertices, either
contains a or contains a . Parsons gave a recursive
formula to determine the values of , where is a path on
vertices and is a star on vertices. In this note, we first
give an explicit formula for the path-star Ramsey numbers. Secondly, we study
the Ramsey numbers , where is a linear forest on
vertices. We determine the exact values of for the cases
and , and for the case that has no odd component.
Moreover, we give a lower bound and an upper bound for the case and has at least one odd component.Comment: 7 page
Rainbow Turán Problems
For a fixed graph H, we define the rainbow Turán number ex^*(n,H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n,H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex^*(n,H)=(1+o(1))ex(n,H), and if H is colour-critical we show that ex^{*}(n,H)=ex(n,H). When H is the complete bipartite graph K_{s,t} with s ≤ t we show ex^*(n,K_{s,t}) = O(n^{2-1/s}), which matches the known bounds for ex(n,K_{s,t}) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex^*(n,C_6) = O(n^{4/3}), which is of the correct order of magnitude
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