352 research outputs found
Induced colorful trees and paths in large chromatic graphs
In a proper vertex coloring of a graph a subgraph is colorful
if its vertices are colored with different colors. It is
well-known (see for example in Gyárfás (1980)) that in every
proper coloring of a k-chromatic graph there is a colorful
path Pk on k vertices. The first author proved in 1987 that
k-chromatic and triangle-free graphs have a path Pk which is
an induced subgraph. N.R. Aravind conjectured that these
results can be put together: in every proper coloring of a k-
chromatic triangle-free graph, there is an induced colorful
Pk. Here we prove the following weaker result providing some
evidence towards this conjecture: For a suitable function
f(k), in any proper coloring of an f(k)-chromatic graph of
girth at least five, there is an induced colorful path on k
vertices
On topological relaxations of chromatic conjectures
There are several famous unsolved conjectures about the chromatic number that
were relaxed and already proven to hold for the fractional chromatic number. We
discuss similar relaxations for the topological lower bound(s) of the chromatic
number. In particular, we prove that such a relaxed version is true for the
Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of
Hadwiger from this point of view. For the latter, a similar statement was
already proven in an earlier paper of the first author with G. Tardos, our main
concern here is that the so-called odd Hadwiger conjecture looks much more
difficult in this respect. We prove that the statement of the odd Hadwiger
conjecture holds for large enough Kneser graphs and Schrijver graphs of any
fixed chromatic number
Induced subgraphs of graphs with large chromatic number. XI. Orientations
Fix an oriented graph H, and let G be a graph with bounded clique number and
very large chromatic number. If we somehow orient its edges, must there be an
induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for
two specific kinds of digraph H: the three-edge path, with the first and last
edges both directed towards the interior; and stars (with many edges directed
out and many directed in). Aboulker et al subsequently conjectured that the
answer is affirmative in both cases. We give affirmative answers to both
questions
Structure and colour in triangle-free graphs
Motivated by a recent conjecture of the first author, we prove that every
properly coloured triangle-free graph of chromatic number contains a
rainbow independent set of size . This is sharp up to
a factor . This result and its short proof have implications for the related
notion of chromatic discrepancy.
Drawing inspiration from both structural and extremal graph theory, we
conjecture that every triangle-free graph of chromatic number contains
an induced cycle of length as . Even if
one only demands an induced path of length , the
conclusion would be sharp up to a constant multiple. We prove it for regular
girth graphs and for girth graphs.
As a common strengthening of the induced paths form of this conjecture and of
Johansson's theorem (1996), we posit the existence of some such that for
every forest on vertices, every triangle-free and induced -free
graph has chromatic number at most . We prove this assertion with
`triangle-free' replaced by `regular girth '.Comment: 12 pages; in v2 one section was removed due to a subtle erro
A survey of -boundedness
If a graph has bounded clique number, and sufficiently large chromatic
number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made
a number of challenging conjectures about this in the early 1980's, which have
remained open until recently; but in the last few years there has been
substantial progress. This is a survey of where we are now
The Parameterised Complexity of List Problems on Graphs of Bounded Treewidth
We consider the parameterised complexity of several list problems on graphs,
with parameter treewidth or pathwidth. In particular, we show that List Edge
Chromatic Number and List Total Chromatic Number are fixed parameter tractable,
parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even
parameterised by pathwidth. These results resolve two open questions of
Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen (2011).Comment: Author final version, to appear in Information and Computation.
Changes from previous version include improved literature references and
restructured proof in Section
A Victorian Age Proof of the Four Color Theorem
In this paper we have investigated some old issues concerning four color map
problem. We have given a general method for constructing counter-examples to
Kempe's proof of the four color theorem and then show that all counterexamples
can be rule out by re-constructing special 2-colored two paths decomposition in
the form of a double-spiral chain of the maximal planar graph. In the second
part of the paper we have given an algorithmic proof of the four color theorem
which is based only on the coloring faces (regions) of a cubic planar maps. Our
algorithmic proof has been given in three steps. The first two steps are the
maximal mono-chromatic and then maximal dichromatic coloring of the faces in
such a way that the resulting uncolored (white) regions of the incomplete
two-colored map induce no odd-cycles so that in the (final) third step four
coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio
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