1,595 research outputs found
Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs
We establish relations between the bandwidth and the treewidth of bounded
degree graphs G, and relate these parameters to the size of a separator of G as
well as the size of an expanding subgraph of G. Our results imply that if one
of these parameters is sublinear in the number of vertices of G then so are all
the others. This implies for example that graphs of fixed genus have sublinear
bandwidth or, more generally, a corresponding result for graphs with any fixed
forbidden minor. As a consequence we establish a simple criterion for
universality for such classes of graphs and show for example that for each
gamma>0 every n-vertex graph with minimum degree ((3/4)+gamma)n contains a copy
of every bounded-degree planar graph on n vertices if n is sufficiently large
Distance-two coloring of sparse graphs
Consider a graph and, for each vertex , a subset
of neighbors of . A -coloring is a coloring of the
elements of so that vertices appearing together in some receive
pairwise distinct colors. An obvious lower bound for the minimum number of
colors in such a coloring is the maximum size of a set , denoted by
. In this paper we study graph classes for which there is a
function , such that for any graph and any , there is a
-coloring using at most colors. It is proved that if
such a function exists for a class , then can be taken to be a linear
function. It is also shown that such classes are precisely the classes having
bounded star chromatic number. We also investigate the list version and the
clique version of this problem, and relate the existence of functions bounding
those parameters to the recently introduced concepts of classes of bounded
expansion and nowhere-dense classes.Comment: 13 pages - revised versio
Dense subgraphs in the H-free process
The H-free process starts with the empty graph on n vertices and adds edges
chosen uniformly at random, one at a time, subject to the condition that no
copy of H is created, where H is some fixed graph. When H is strictly
2-balanced, we show that for some c,d>0, with high probability as , the final graph of the H-free process contains no subgraphs F on vertices with maximum density . This extends and generalizes results of Gerke and Makai for the C_3-free
process.Comment: 7 pages, revised versio
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