Edge coloring multigraphs without small dense subsets

© 2015. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/One consequence of a long-standing conjecture of Goldberg and Seymour about the chromatic index of multigraphs would be the following statement. Suppose $G$ is a multigraph with maximum degree $\Delta$, such that no vertex subset $S$ of odd size at most $\Delta$ induces more than $(\Delta+1)(|S|-1)/2$ edges. Then $G$ has an edge coloring with $\Delta+1$ colors. Here we prove a weakened version of this statement.Natural Sciences and Engineering Research Counci

A note on intersecting hypergraphs with large cover number

Published by 'The Electronic Journal of Combinatorics' at 10.37236/6460.We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r−4 for all but finitely many r. This answers a question of Abu-Khazneh, Barát, Pokrovskiy and Szabó, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r.Natural Sciences and Engineering Research Council of Canad

Partitioning by Monochromatic Trees

AbstractAnyr-edge-colouredn-vertex complete graphKncontains at mostrmonochromatic trees, all of different colours, whose vertex sets partition the vertex set ofKn, providedn⩾3r4r! (1−1/r)3(1−r)logr. This comes close to proving, for largen, a conjecture of Erdős, Gyárfás, and Pyber, which states thatr−1 trees suffice for alln

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

External and Ramsey type results for graphs and hypergraphs

SIGLEAvailable from British Library Document Supply Centre-DSC:D063124 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Edge coloring multigraphs without small dense subsets

One consequence of an old conjecture of Goldberg and Seymour about the chromatic index of multigraphs would be the following statement. Suppose G is a multigraph with maximum degree ∆, such that no vertex subset S of odd size at most ∆ induces more than (∆+1)(|S|−1)/2 edges. Then G has an edge coloring with ∆ + 1 colors. Here we prove a weakened version of this statement.