238 research outputs found
Local And Global Colorability of Graphs
It is shown that for any fixed and , the maximum possible
chromatic number of a graph on vertices in which every subgraph of radius
at most is colorable is (that is, up to a factor poly-logarithmic in ).
The proof is based on a careful analysis of the local and global colorability
of random graphs and implies, in particular, that a random -vertex graph
with the right edge probability has typically a chromatic number as above and
yet most balls of radius in it are -degenerate
Counting triangles in some Ramsey graphs
We extend Goodman’s result on the cardinality of monochromatic triangles in a 2-colored complete graph to the case of bounding the number of triangles in the first color. We apply it to derive the upper bounds on some non-diagonal Ramsey numbers. In particular we show that R(K4 − e,K8) \u3c= 45
Threshold phenomena in random graphs
In the 1950s, random graphs appeared for the first time in a result of the prolific hungarian mathematician Pál Erd\H{o}s. Since then, interest in random graph theory has only grown up until now. In its first stages, the basis of its theory were set, while they were mainly used in probability and combinatorics theory. However, with the new century and the boom of technologies like the World Wide Web, random graphs are even more important since they are extremely useful to handle problems in fields like network and communication theory. Because of this fact, nowadays random graphs are widely studied by the mathematical community around the world and new promising results have been recently achieved, showing an exciting future for this field. In this bachelor thesis, we focus our study on the threshold phenomena for graph properties within random graphs
On the Spectrum of Wenger Graphs
Let , where is a prime and is an integer. For ,
let and be two copies of the -dimensional vector spaces over the
finite field . Consider the bipartite graph with partite
sets and defined as follows: a point is adjacent to a line if and only if the
following equalities hold: for . We call the graphs Wenger graphs. In this paper, we determine all
distinct eigenvalues of the adjacency matrix of and their
multiplicities. We also survey results on Wenger graphs.Comment: 9 pages; accepted for publication to J. Combin. Theory, Series
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