982 research outputs found
Non-three-colorable common graphs exist
A graph H is called common if the total number of copies of H in every graph
and its complement asymptotically minimizes for random graphs. A former
conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that
every graph is common. Thomason disproved both conjectures by showing that the
complete graph of order four is not common. It is now known that in fact the
common graphs are very rare. Answering a question of Sidorenko and of Jagger,
Stovicek and Thomason from 1996 we show that the 5-wheel is common. This
provides the first example of a common graph that is not three-colorable.Comment: 9 page
Characterising and recognising game-perfect graphs
Consider a vertex colouring game played on a simple graph with
permissible colours. Two players, a maker and a breaker, take turns to colour
an uncoloured vertex such that adjacent vertices receive different colours. The
game ends once the graph is fully coloured, in which case the maker wins, or
the graph can no longer be fully coloured, in which case the breaker wins. In
the game , the breaker makes the first move. Our main focus is on the
class of -perfect graphs: graphs such that for every induced subgraph ,
the game played on admits a winning strategy for the maker with only
colours, where denotes the clique number of .
Complementing analogous results for other variations of the game, we
characterise -perfect graphs in two ways, by forbidden induced subgraphs
and by explicit structural descriptions. We also present a clique module
decomposition, which may be of independent interest, that allows us to
efficiently recognise -perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the
International Colloquium on Graph Theory (ICGT) 201
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
Graphs that are not pairwise compatible: A new proof technique (extended abstract)
A graph G = (V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dminand dmax, dmin≤ dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin≤ dT(u, v) ≤ dmax, where dT(u, v) is the sum of the weights of the edges on the unique path PT(u, v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. C28), so improving the previously known result concerning the smallest planar graph known not to be PCG
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