5 research outputs found
On graphs with a large chromatic number containing no small odd cycles
In this paper, we present the lower bounds for the number of vertices in a
graph with a large chromatic number containing no small odd cycles
Two results on the digraph chromatic number
It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there
exist graphs of maximum degree and of arbitrarily large girth whose
chromatic number is at least . We show an analogous
result for digraphs where the chromatic number of a digraph is defined as
the minimum integer so that can be partitioned into acyclic
sets, and the girth is the length of the shortest cycle in the corresponding
undirected graph. It is also shown, in the same vein as an old result of Erdos
(1962), that there are digraphs with arbitrarily large chromatic number where
every large subset of vertices is 2-colorable
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