arXiv subjects
Publications and source records attributed to Daniel Kral.
We show that every cubic bridgeless graph G has at least 2^(|V(G)|/3656) perfect matchings. This confirms an old conjecture of Lovasz and Plummer. This version of the paper uses a different definition of a burl from the journal version of the paper and a different proof of Lemma 18 is given. This simplifies the exposition of our arguments throughout the whole paper.
Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order logic properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.
Let m(G) be the maximum number of vertex-disjoint odd cycles of a graph G and t(G) the minimum number of vertices whose removal makes G bipartite. We show that t(G)<=6m(G) if G is planar. This improves the previous bound t(G)<=10m(G) by Fiorini, Hardy, Reed and Vetta [Math. Program. Ser. B 110 (2007), 71-91].
A class of graphs G is chi-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is chi-bounded,then every class of graphs admitting a decomposition along cuts of small rank to graphs from G is chi-bounded. As a corollary, we obtain that every class of graphs with bounded rank-width (or equivalently, clique-width) is chi-bounded.
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.
We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.
Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1.
Reed conjectured that for every $\varepsilon>0$ and every integer $Δ$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $Δ$ and girth at least $g$ is at most $Δ+1+\varepsilon$. The conjecture was proven to be true when $Δ=3$ or $Δ$ is even. We settle the conjecture by proving it for the remaining cases.
The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with $m$ edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m/21\approx 1.619m$ and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m/27\approx 1.630m$.
We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n/g)<3n/10+O(n/g).
We introduce a new width parameter for matroids called decomposition width and prove that every matroid property expressible in the monadic second order logic can be computed in linear time for matroids with bounded decomposition width if their decomposition is given. Since decompositions of small width for our new notion can be computed in polynomial time for matroids of bounded branch-width represented over finite fields, our results include recent algorithmic results of Hlineny [J. Combin. Theory Ser. B 96 (2006), 325-351] in this area and extend his results to matroids not necessarily representable over finite fields.
We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertices. Improving an upper bound of Axenovich and Iverson, we show that maxR(n,K_m,K_4) <= n^{3/2}\sqrt{2m} for all m >= 3. Further, we discuss a possible way to improve their lower bound on maxR(n,K_4,K_4) based on incidence graphs of finite projective planes.
We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.