6,683 research outputs found
A note on chromatic properties of threshold graphs
In threshold graphs one may find weights for the vertices and a threshold value t such that for any subset S of vertices, the sum of the weights is at most the threshold t if and only if the set S is a stable (independent) set. In this note we ask a similar question about vertex colorings: given an integer p, when is it possible to find weights (in general depending on p) for the vertices and a threshold value tp such that for any subset S of vertices the sum of the weights is at most tp if and only if S generates a subgraph with chromatic number at most p − 1? We show that threshold graphs do have this property and we show that one can even find weights which are valid for all values of p simultaneously
A note on circular chromatic number of graphs with large girth and similar problems
In this short note, we extend the result of Galluccio, Goddyn, and Hell,
which states that graphs of large girth excluding a minor are nearly bipartite.
We also prove a similar result for the oriented chromatic number, from which
follows in particular that graphs of large girth excluding a minor have
oriented chromatic number at most , and for the th chromatic number
, from which follows in particular that graphs of large girth
excluding a minor have
Chromatic thresholds in dense random graphs
The chromatic threshold of a graph with respect to the
random graph is the infimum over such that the following holds
with high probability: the family of -free graphs with
minimum degree has bounded chromatic number. The study of
the parameter was initiated in 1973 by
Erd\H{o}s and Simonovits, and was recently determined for all graphs . In
this paper we show that for all fixed , but that typically if . We also make significant progress towards determining
for all graphs in the range . In sparser random graphs the
problem is somewhat more complicated, and is studied in a separate paper.Comment: 36 pages (including appendix), 1 figure; the appendix is copied with
minor modifications from arXiv:1108.1746 for a self-contained proof of a
technical lemma; accepted to Random Structures and Algorithm
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