792 research outputs found
On bounding the difference between the maximum degree and the chromatic number by a constant
We provide a finite forbidden induced subgraph characterization for the graph
class , for all , which is defined as
follows. A graph is in if for any induced subgraph, holds, where is the maximum degree and is the
chromatic number of the subgraph.
We compare these results with those given in [O. Schaudt, V. Weil, On
bounding the difference between the maximum degree and the clique number,
Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI:
10.1007/s00373-014-1468-3], where we studied the graph class , for
, whose graphs are such that for any induced subgraph,
holds, where denotes the clique number of
a graph. In particular, we give a characterization in terms of
and of those graphs where the neighborhood of every vertex is
perfect.Comment: 10 pages, 4 figure
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
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