Julia Ehrenmüller

A conjecture by Aharoni and Berger states that every family of $n$ matchings of size $n+1$ in a bipartite multigraph contains a rainbow matching of size $n$. In this paper we prove that matching sizes of $\left(\frac 3 2 + o(1)\right) n$ suffice to guarantee such a rainbow matching, which is asymptotically the same bound as the best known one in case we only aim to find a rainbow matching of size $n-1$. This improves previous results by Aharoni, Charbit and Howard, and Kotlar and Ziv.

By means of analytic techniques we show that the expected number of spanning trees in a connected labelled series-parallel graph on $n$ vertices chosen uniformly at random satisfies an estimate of the form $s \varrho^{-n} (1+o(1))$, where $s$ and $\varrho$ are computable constants, the values of which are approximately $s \approx 0.09063$ and $\varrho^{-1} \approx 2.08415$. We obtain analogue results for subfamilies of series-parallel graphs including 2-connected series-parallel graphs, 2-trees, and series-parallel graphs with fixed excess.

Motivated by a question of Grinblat, we study the minimal number $\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are equivalence relations on a set $X$ such that for every $i\in[n]$ there are at least $\mathfrak{v}(n)$ elements whose equivalence classes with respect to $A_i$ are nontrivial, then $A_1, \ldots, A_n$ contain a rainbow matching, i.e. there exist $2n$ distinct elements $x_1,y_1,\ldots,x_n,y_n\in X$ with $x_i\sim_{A_i} y_i$ for each $i\in [n]$. Grinblat asked whether $\mathfrak{v}(n) = 3n-2$ for every $n\geq 4$. The best-known upper bound was $\mathfrak{v}(n) \leq 16n/5 + \mathcal{O}(1)$ due to Nivash and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that $\mathfrak{v}(n) = 3n+o(n)$.

In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series-parallel if it does not contain $K_4$ as a minor. Series-parallel graphs are also known as partial 2-trees, which are arbitrary subgraphs of 2-trees. We present a proof that every connected series-parallel graph has a vertex that is common to all of its longest paths. Since 2-trees are maximal series-parallel graphs, and outerplanar graphs are also series-parallel, our result captures these two classes in one proof and strengthens them to a larger class of graphs. We also describe how this vertex can be found in linear time.

A conjecture by Aharoni and Berger states that every family of $n$ matchings of size $n+1$ in a bipartite multigraph contains a rainbow matching of size $n$. In this paper we prove that matching sizes of $(3/2 + o(1)) n$ suffice to guarantee such a rainbow matching, which is asymptotically the same bound as the best known one in case we only aim to find a rainbow matching of size $n-1$. This improves previous results by Aharoni, Charbit and Howard, and Kotlar and Ziv.

We consider biased $(1:b)$ Avoider-Enforcer games in the monotone and strict versions. In particular, we show that Avoider can keep his graph being a forest for every but maybe the last round of the game if $b \geq 200 n \ln n$. By this we obtain essentially optimal upper bounds on the threshold biases for the non-planarity game, the non-$k$-colorability game, and the $K_t$-minor game thus addressing a question and improving the results of Hefetz, Krivelevich, Stojaković, and Szabó. Moreover, we give a slight improvement for the lower bound in the non-planarity game.

We consider biased $(1:b)$ Avoider-Enforcer games in the monotone and strict versions. In particular, we show that Avoider can keep his graph being a forest for every but maybe the last round of the game if $b \geq 200 n \ln n$. By this we obtain essentially optimal upper bounds on the threshold biases for the non-planarity game, the non-$k$-colorability game, and the $K_t$-minor game thus addressing a question and improving the results of Hefetz, Krivelevich, Stojakovi\'c, and Szab\'o. Moreover, we give a slight improvement for the lower bound in the non-planarity game.