Related papers: The biased odd cycle game

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases…

We introduce and analyze the Walker-Breaker game, a variant of Maker-Breaker games where Maker is constrained to choose edges of a walk or path in a given graph G, with the goal of visiting as many vertices of the underlying graph as…

We consider a game played on an initially empty graph where two players alternate drawing an edge between vertices subject to the condition that no degree can exceed $k$. We show that for $k=3$, either player can avoid a Hamilton cycle, and…

We initiate the study of a new variant of the Maker-Breaker positional game, which we call multistage game. Given a hypergraph $\mathcal{H}=(\mathcal{X},\mathcal{F})$ and a bias $b \ge 1$, the $(1:b)$ multistage Maker-Breaker game on…

Given a fixed graph $H$ with at least two edges and positive integers $n$ and $b$, the strict $(1 \colon b)$ Avoider-Enforcer $H$-game, played on the edge set of $K_n$, has the following rules: In each turn Avoider picks exactly one edge,…

Let \mathcal{F}_k denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollob\'as conjectured that for every \epsilon>0 and positive…

We study the $(p,q)$-Maker Breaker Crossing game introduced by Day and Falgas Ravry in 'Maker-Breaker percolation games I: crossing grids'. The game described in their paper involves two players Maker and Breaker who take turns claiming p…

In the Maker-Breaker resolving game, two players named Resolver and Spoiler alternately select unplayed vertices of a given graph $G$. The aim of Resolver is to select all the vertices of some resolving set of $G$, while Spoiler aims to…

A predominated graph is a pair $(G,D)$, where $G$ is a graph and the vertices in $D\subseteq V(G)$ are considered already dominated. Maker-Breaker domination game critical (MBD critical) predominated graphs are introduced as the…

In this short note we prove that for every $k\in \mathbb{N}$ there is a $t_k\in\mathbb{N}$ such that for every digraph $G$ there are either $k$ edge-disjoint directed cycles in $G$ or a set $X$ of at most $t_k$ edges such that $G-X$…

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-H\"{a}ggkvist Conjecture, we study the existence of properly colored cycles of bounded length…

Consider a graph with $n$ vertices where the shortest odd cycle is of length $>2k+1$. We revisit two known results about such graphs: (I) Such a graph is almost bipartite, in the sense that it can be made bipartite by removing from it…

Let $\Lambda$ be an infinite connected graph, and let $v_0$ be a vertex of $\Lambda$. We consider the following positional game. Two players, Maker and Breaker, play in alternating turns. Initially all edges of $\Lambda$ are marked as…

We introduce and study two Maker-Breaker-like games for constructing planar graphs: the edge drawing game, where two players take turns drawing non-intersecting edges between points in the plane, and the circle packing game, where the…

A classic theorem of Erd\H{o}s and P\'osa (1965) states that every graph has either $k$ vertex-disjoint cycles or a set of $O(k \log k)$ vertices meeting all its cycles. While the standard proof revolves around finding a large `frame' in…

We prove that the set of long cycles has the edge-Erd\H{o}s-P\'osa property: for every fixed integer $\ell\ge 3$ and every $k\in\mathbb{N}$, every graph $G$ either contains $k$ edge-disjoint cycles of length at least $\ell$ (long cycles) or…

We say that two graphs $H_1,H_2$ on the same vertex set are $G$-creating ($G$-different in other papers, this difference is explained in the introduction) if the union of the two graphs contains $G$ as a subgraph. Let $H(n,k)$ be the…

In an influential paper, Erd\H{o}s and Selfridge introduced the Maker-Breaker game played on a hypergraph, or equivalently, on a monotone CNF. The players take turns assigning values to variables of their choosing, and Breaker's goal is to…

A biclique of a graph $G$ is a maximal induced complete bipartite subgraph of $G$. The biclique graph of $G$ denoted by $KB(G)$, is the intersection graph of all the bicliques of $G$. The biclique graph can be thought as an operator between…

Resolving a conjecture of Bollob\'{a}s and Erd\H{o}s, Gy\'{a}rf\'{a}s proved that every graph $G$ of chromatic number $k+1\geq 3$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ distinct odd lengths. We strengthen this prominent result by…