4,520 research outputs found
Partitions of graphs into cographs
AbstractCographs form the minimal family of graphs containing K1 that is closed with respect to complementation and disjoint union. We discuss vertex partitions of graphs into the smallest number of cographs. We introduce a new parameter, calling the minimum order of such a partition the c-chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c-chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c-chromatic number and twice this number. We show that both bounds are sharp for graphs with arbitrarily high girth. This provides an alternative proof to a result by Broere and Mynhardt; namely, there exist triangle-free graphs with arbitrarily large c-chromatic numbers. We show that any planar graph with girth at least 11 has a c-chromatic number at most two. We close with several remarks on computational complexity. In particular, we show that computing the c-chromatic number is NP-complete for planar graphs
Bipartite induced density in triangle-free graphs
We prove that any triangle-free graph on vertices with minimum degree at
least contains a bipartite induced subgraph of minimum degree at least
. This is sharp up to a logarithmic factor in . Relatedly, we show
that the fractional chromatic number of any such triangle-free graph is at most
the minimum of and as . This is
sharp up to constant factors. Similarly, we show that the list chromatic number
of any such triangle-free graph is at most as
.
Relatedly, we also make two conjectures. First, any triangle-free graph on
vertices has fractional chromatic number at most
as . Second, any triangle-free
graph on vertices has list chromatic number at most as
.Comment: 20 pages; in v2 added note of concurrent work and one reference; in
v3 added more notes of ensuing work and a result towards one of the
conjectures (for list colouring
Some results on chromatic number as a function of triangle count
A variety of powerful extremal results have been shown for the chromatic
number of triangle-free graphs. Three noteworthy bounds are in terms of the
number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994),
and Johansson. There have been comparatively fewer works extending these types
of bounds to graphs with a small number of triangles. One noteworthy exception
is a result of Alon et. al (1999) bounding the chromatic number for graphs with
low degree and few triangles per vertex; this bound is nearly the same as for
triangle-free graphs. This type of parametrization is much less rigid, and has
appeared in dozens of combinatorial constructions.
In this paper, we show a similar type of result for as a function
of the number of vertices , the number of edges , as well as the triangle
count (both local and global measures). Our results smoothly interpolate
between the generic bounds true for all graphs and bounds for triangle-free
graphs. Our results are tight for most of these cases; we show how an open
problem regarding fractional chromatic number and degeneracy in triangle-free
graphs can resolve the small remaining gap in our bounds
- …