38 research outputs found
Cuts in matchings of 3-connected cubic graphs
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette,
Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and
on even graphs in digraphs whose contraction is strongly connected
(Hochst\"attler). We show that all of them fit into the same framework related
to cuts in matchings. This allows us to find a counterexample to the conjecture
of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all
planar graphs on at most 26 vertices. Finally, we state a new conjecture on
bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio
Planar digraphs without large acyclic sets
Given a directed graph, an acyclic set is a set of vertices inducing a
subgraph with no directed cycle. In this note we show that there exist oriented
planar graphs of order for which the size of the maximum acyclic set is at
most , for any . This disproves a conjecture of
Harutyunyan and shows that a question of Albertson is best possible.Comment: 3 pages, 1 figur
On powers of interval graphs and their orders
It was proved by Raychaudhuri in 1987 that if a graph power is an
interval graph, then so is the next power . This result was extended to
-trapezoid graphs by Flotow in 1995. We extend the statement for interval
graphs by showing that any interval representation of can be extended
to an interval representation of that induces the same left endpoint and
right endpoint orders. The same holds for unit interval graphs. We also show
that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the
main result of this note, follows from earlier results of [G. Agnarsson, P.
Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and
dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003].
This version is updated accordingl
Strong edge colouring of subcubic graphs
AbstractA strong edge colouring of a graph G is a proper edge colouring such that every path of length 3 uses three colours. In this paper, we prove that every subcubic graph with maximum average degree strictly less than 157 (resp. 2711, 135, 3613) can be strong edge coloured with six (resp. seven, eight, nine) colours
Strong edge-colouring of sparse planar graphs
A strong edge-colouring of a graph is a proper edge-colouring where each
colour class induces a matching. It is known that every planar graph with
maximum degree has a strong edge-colouring with at most
colours. We show that colours suffice if the graph has girth 6, and
colours suffice if or the girth is at least 5. In the
last part of the paper, we raise some questions related to a long-standing
conjecture of Vizing on proper edge-colouring of planar graphs
Enumerating -arc-connected orientations
12 pagesWe study the problem of enumerating the -arc-connected orientations of a graph , i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with delay and amortized time , which improves over the analysis of the submodular flow algorithm. As ingredients, we obtain enumeration algorithms for the -orientations of a graph in delay and for the outdegree sequences attained by -arc-connected orientations of in delay
Feedback vertex sets in (directed) graphs of bounded degeneracy or treewidth
We study the minimum size of a feedback vertex set in directed and
undirected -vertex graphs of given degeneracy or treewidth. In the
undirected setting the bound is known to be tight for graphs
with bounded treewidth or bounded odd degeneracy . We show that neither
of the easy upper and lower bounds and can
be exact for the case of even degeneracy. More precisely, for even degeneracy
we prove that , there exists
a -degenerate graph for which .
For directed graphs of bounded degeneracy , we prove that
and that this inequality is strict when is odd. For
directed graphs of bounded treewidth , we show that and for every , there exists a -degenerate graph
for which . Further,
we provide several constructions of low degeneracy or treewidth and large .Comment: 19 pages, 7 figures, 2 table
Strong edge-coloring of -bipartite graphs
A strong edge-coloring of a graph is an assignment of colors to edges
such that every color class induces a matching. We here focus on bipartite
graphs whose one part is of maximum degree at most and the other part is of
maximum degree . For every such graph, we prove that a strong
-edge-coloring can always be obtained. Together with a result of
Steger and Yu, this result confirms a conjecture of Faudree, Gy\'arf\'as,
Schelp and Tuza for this class of graphs