2 research outputs found
Constrained Ramsey Numbers
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum
n such that every edge coloring of the complete graph on n vertices, with any
number of colors, has a monochromatic subgraph isomorphic to S or a rainbow
(all edges differently colored) subgraph isomorphic to T. The Erdos-Rado
Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star
or T is acyclic, and much work has been done to determine the rate of growth of
f(S, T) for various types of parameters. When S and T are both trees having s
and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <=
O(st^2) and conjectured that it is always at most O(st). They also mentioned
that one of the most interesting open special cases is when T is a path. In
this work, we study this case and show that f(S, P_t) = O(st log t), which
differs only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision
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Cliques in graphs
The main focus of this thesis is to evaluate , the minimal number of -cliques in graphs with vertices and minimum degree~. A fundamental result in Graph Theory states that a triangle-free graph of order has at most edges. Hence, a triangle-free graph has minimum degree at most , so if then . For , I have evaluated and determined the structures of the extremal graphs. For , I give a conjecture on , as well as the structures of these extremal graphs. Moreover, I have proved various partial results that support this conjecture. Let be the analogous version of for regular graphs. Notice that there exist and such that but . For example, a theorem of Andr{\'a}sfai, Erd{\H{o}}s and S{\'o}s states that any triangle-free graph of order with minimum degree greater than must be bipartite. Hence but for odd. I have evaluated the exact value for between and and determined the structure of these extremal graphs.
At the end of the thesis, I investigate a question in Ramsey Theory. The Ramsey number of a graph is the minimum number , such that any edge colouring of with colours contains a monochromatic copy of . The constrained Ramsey number of two graphs and is the minimum number such that any edge colouring of with any number of colours contains a monochromatic copy of or a rainbow copy of . It turns out that these two quantities are closely related when is a matching. Namely, for almost all graphs , for