Abstract:
A sequence $s_1,s_2,...,s_k,s_1,s_2,...,s_k$ is a repetition. A sequence $S$ is nonrepetitive, if no subsequence of consecutive terms of $S$ form a repetition. Let $G$ be a vertex colored graph. A path of $G$ is nonrepetitive, if the sequence of colors on its vertices is nonrepetitive. If $G$ is a plane graph, then a facial nonrepetitive vertex coloring of $G$ is a vertex coloring such that any facial path is nonrepetitive. Let $\pi_f(G)$ denote the minimum number of colors of a facial nonrepetitive vertex coloring of $G$. Jendro\vl and Harant posed a conjecture that $\pi_f(G)$ can be bounded from above by a constant. We prove that $\pi_f(G)\le 24$ for any plane graph $G$.

Abstract:
A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path $v_1,v_2,...,v_{2t}$ such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree $\Delta$ has a $f(\Delta)$-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree $\Delta$ has a $O(k\Delta)$-colouring that is nonrepetitive on paths, and a $O(k\Delta^3)$-colouring that is nonrepetitive on walks.

Abstract:
We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just 6 short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-5}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.

Abstract:
We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Bar\'at and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a 191-edge-connected graph of size divisible by 4, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star.

Abstract:
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only $\frac{45}{17}n + O(1)\approx 2.647n$ edges and maximal 1-plane graphs with only $\frac{7}{3}n+O(1)\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\frac{28}{13}n-O(1)\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges. We improve both lower bounds to $\frac{20n}{9}\approx 2.22n$.

Abstract:
Albertson conjectured that if a graph $G$ has chromatic number $r$ then its crossing number is at least as much as the crossing number of $K_r$. Albertson, Cranston, and Fox verified the conjecture for $r\le 12$. We prove the statement for $r\le 16$.

Abstract:
A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}$. We also improve the general lower bound on $f(r)$, showing that $f(r)\ge 3.052r+O(1)$. We show that a stronger form of Ryser's conjecture that was used to prove the $r=3$ case fails for all $r>3$. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an $r$-partite hypergraph there exists a set $S$ of size at most $r-1$, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.

Abstract:
The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.

Abstract:
For a property $\Gamma$ and a family of sets $\cF$, let $f(\cF,\Gamma)$ be the size of the largest subfamily of $\cF$ having property $\Gamma$. For a positive integer $m$, let $f(m,\Gamma)$ be the minimum of $f(\cF,\Gamma)$ over all families of size $m$. A family $\cF$ is said to be $B_d$-free if it has no subfamily $\cF'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold. A family $\cF$ is $a$-union free if $F_1\cup ... F_a \neq F_{a+1}$ whenever $F_1,..,F_{a+1}$ are distinct sets in $\FF$. We verify a conjecture of Erd\H os and Shelah that $f(m, B_2\text{\rm -free})=\Theta(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union free})$.

Abstract:
Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with path-width at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight to structural properties. We take a different route, and concentrate on the building blocks and how they are glued together. In this way, we get a characterization of 2-connected and 2-edge-connected graphs with path-width at most two. Along similar lines, we sketch the complete characterization of graphs with path-width at most two.