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Nathan Bowler

Publications and source records attributed to Nathan Bowler.

At least 19 recordsLinked to original sources
arXiv2026

Maker-Breaker games on infinite graphs with precolored edges

Suppose we are given graphs $B$ and $G$. In the classical Maker-Breaker game $\text{MB}(B,G)$ two players, Maker and Breaker, alternately claim edges of $B$ and it is Maker's goal to claim a copy of $G$ in $B$, while it is Breaker's goal to prevent that. In this paper, $B$ is the countably infinite complete graph $K_{\aleph_0}$ and we are given finitely many infinite subgraphs $G_1, \dots, G_k \subseteq B$. In the color preserving game, it will be Maker's goal to claim a $K_{\aleph_0} \subseteq B$, which contains infinitely many edges of each $G_i$. We present sufficient winning conditions for both Maker and Breaker, if $k > 1$ and a full characterization of the game, if $k =1$. This partly answers a question of Bowler, Emde and Gut. In the (partially) pattern preserving game, it is Maker's goal to claim a copy $K$ of $K_{\aleph_0}$, such that $G_i \cap K$ is isomorphic to (a subgraph of) $G_i$ for all $i \in [k]$. In those games, we investigate some patterns for which Maker has a winning strategy.

math.CO↗
arXiv2026

Torsoids in Path-Like Graphs without nontrivial even 2-separation

Bowler et al. introduced the concept of torsoids that describes the 1-separations of a directed graph in a canonical way building on previous progress by Lov{\'a}sz. To fully understand the structure of directed graphs with respect to their 1-separations it remains to find a global structure along which the torsoids are arranged. In this paper, we start the investigation of this global structure for a specific class of directed graphs.

math.CO↗
arXiv2026

On the Possibilities of Defining Infinite Oriented Matroids

Is it possible to define cryptomorphic axiom systems for infinite oriented matroids by lifting some of the axiom systems for finite oriented matroids to the infinite setting while not losing duality in the process? We show that the answer to this question is a twofold "no". First, lifting the circuit axioms neither preserves duality nor inheritance of strong circuit elimination in minors. Second, although duality is kept intact by translating the orthogonality axioms and an axiom system based on the Farkas Lemma, the classes of infinite oriented matroids obtained in this way have the property that one is a proper subclass of the other.

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arXiv2025

Locally interval graphs are circular-arc graphs

Circular-arc graphs are graphs that can be represented as intersection graphs of subpaths of a cycle. Interval graphs are graphs that can be represented as intersection graphs of subpaths of a path. Since cycles are locally paths, every circular-arc graph is locally interval. In this paper, we prove that the converse holds as well: every locally interval graph is a circular-arc graph. This result and its proofs are connected to a recent broader study of structural local-global theory and build on previous work on locally chordal graphs.

math.CO↗
arXiv2025

Drawing strategies in Strong Ramsey games for 3-uniform hypergraphs

The Strong Ramsey game $\mathcal{R}(B,G)$ is a two player game with players $P_1$ and $P_2$, where $B$ and $G$ are $k$-uniform hypergraphs for some $k \geq 2$. $G$ is always finite, while $B$ may be infinite. $P_1$ and $P_2$ alternately color uncolored edges $e \in B$ in their respective color and $P_1$ begins. Whoever completes a monochromatic copy of $G$ in their own color first, wins the game. If no one claims a monochromatic copy of $G$ in a finite number of moves, the game is declared a draw. In this paper, we give an infinite set of 3-uniform hypergraphs $\{G_t\}_{t \geq 3}$, such that $P_2$ has a drawing strategy in the Strong Ramsey game $\mathcal{R}(K_{\aleph_0}^{(3)}, G_t)$. This improves a result by David, Hartarsky and Tiba.

math.CO↗
arXiv2025

$\mathcal{R}(K_{\aleph_0}, \hat{K}_{2,3})$ is a win for Player 1

The Strong Ramsey game $\mathcal{R}(B,G)$ is a two player game with players $P_1$ and $P_2$, where $B$ and $G$ are $k$-uniform hypergraphs for some $k \geq 2$. $G$ is always finite, while $B$ may be infinite. $P_1$ and $P_2$ alternately color uncolored edges $e \in B$ in their respective color and $P_1$ begins. Whoever completes a monochromatic copy of $G$ in their own color first, wins the game. If no one claims a monochromatic copy of $G$ in a finite number of moves, the game is declared a draw. For a $t \in \mathbb{N}$, let $\hat{K}_{2,t}$ denote the $K_{2,t}$ together with the edge connecting the two vertices in the partition class of size 2. The purpose of this paper is to give a winning strategy for $P_1$ in the game $\mathcal{R}(K_{\aleph_0}, \hat{K}_{2,3})$.

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arXiv2025

A structure theorem for rooted connectivity in bidirected graphs

Recently, bidirected graphs have received increasing attention from the graph theory community with both structural and algorithmic results. Bidirected graphs are a generalization of directed graphs, consisting of an undirected graph together with a map assigning each endpoint of every edge either sign $+$ or $-$. The connectivity properties of bidirected graphs are more complex than those of directed graphs and not yet well understood. In this paper, we show a structure theorem about rooted connectivity in bidirected graphs in terms of directed graphs. As applications, we prove Lov\'asz' flame theorem, Pym's theorem and a strong variant of Menger's theorem for a class of bidirected graphs and provide counterexamples in the general case.

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arXiv2025

Probabilistic Strategies: Definability and the Tensor Completeness Problem

Programs that combine I/O and countable probabilistic choice, modulo either bisimilarity or trace equivalence, can be seen as describing a probabilistic strategy. For well-founded programs, we might expect to axiomatize bisimilarity via a sum of equational theories and trace equivalence via a tensor of such theories. This is by analogy with similar results for nondeterminism, established previously. While bisimilarity is indeed axiomatized via a sum of theories, and the tensor is indeed at least sound for trace equivalence, completeness in general, remains an open problem. Nevertheless, we show completeness in the case that either the probabilistic choice or the I/O operations used are finitary. We also show completeness up to impersonation, i.e. that the tensor theory regards trace equivalent programs as solving the same system of equations. This entails completeness up to the cancellation law of the probabilistic choice operator. Furthermore, we show that a probabilistic trace strategy arises as the semantics of a well-founded program iff it is victorious. This means that, when the strategy is played against any partial counterstrategy, the probability of play continuing forever is zero. We link our results (and open problem) to particular monads that can be used to model computational effects.

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arXiv2024

Hitting cycles through prescribed vertices or edges

We prove that for every set $S$ of vertices of a directed graph $D$, the maximum number of vertices in $S$ contained in a collection of vertex-disjoint cycles in $D$ is at least the minimum size of a set of vertices that hits all cycles containing a vertex of $S$. As a consequence, the directed tree-width of a directed graph is linearly bounded in its cycle-width, which improves the previously known quadratic upper bound. We further show that the corresponding statement in bidirected graphs is true and that its edge-variant holds in both undirected and directed graphs, but fails in bidirected graphs. The vertex-version in undirected graphs remains an open problem.

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arXiv2024

Circuit-partition of infinite matroids

Komj\'ath, Milner, and Polat investigated when a finitary matroid admits a partition into circuits. They defined the class of ``finite matching extendable'' matroids and showed in their compactness theorem that those matroids always admit such a partition. Their proof is based on Shelah's singular compactness technique and a careful analysis of certain $\triangle$-systems. We provide a short, simple proof of their theorem. Then we show that a finitary binary oriented matroid can be partitioned into directed circuits if and only if, in every cocircuit, the cardinality of the negative and positive edges is the same. This generalizes an earlier conjecture of Thomassen, settled affirmatively by the second author, about partitioning the edges of an infinite directed graph into directed cycles. As side results, a Laviolette theorem for finitary matroids and a Farkas lemma for finitary binary oriented matroids are proven. An example is given to show that, in contrast to finite oriented matroids, `binary' is essential in the latter result.

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arXiv2024

Connectoids II: existence of normal trees

In this series, we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects such as undirected graphs, directed graphs, bidirected graphs, hypergraphs and finitary matroids. In the first paper, we developed a universal end space theory based on connectoids that unifies the existing end spaces of undirected and directed graphs. In this paper, we establish normal trees of connectoids as a natural generalisation of normal trees of undirected graphs, which are one of the most important tools in infinite graph theory. More precisely, we show that the existence of normal trees of connectoids can be characterised in the same way as for normal trees of undirected graphs: We extend Jung's famous characterisation via dispersed sets to connectoids, and prove that normal spanning trees exist if they exist in some neighbourhood of each end. Furthermore, we show that a connectoid has a normal spanning tree if and only if its groundset can be well-ordered in a certain way, called countable separation number.

math.CO↗
arXiv2024

Connectoids I: a universal end space theory

In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids. In this paper we develop a universal end space theory based on connectoids: the end spaces of connectoids unify the existing end spaces of undirected and directed graphs and establish end spaces for bidirected graphs, hypergraphs and finitary matroids. The main result shows that the tangle-like description of ends in undirected graphs, called directions, extends to connectoids: there is a one-to-one correspondence between the directions of a connectoid and its ends. Furthermore, we generalise normal trees of undirected graphs to connectoids and show that normal trees represent the ends of a connectoid as they do for undirected graphs.

math.CO↗
arXiv2023

The Rational Number Game

We investigate a game played between two players, Maker and Breaker, on a countably infinite complete graph where the vertices are the rational numbers. The players alternately claim unclaimed edges. It is Maker's goal to have after countably many turns a complete infinite graph contained in her coloured edges where the vertex set of the subgraph is order-isomorphic to the rationals. It is Breaker's goal to prevent Maker from achieving this. We prove that there is a winning strategy for Maker in this game. We also prove that there is a winning strategy for Breaker in the game where Maker must additionally make the vertex set of her complete graph dense in the rational numbers.

math.CO↗
arXiv2023

The $K^4$-Game

We investigate a two player game called the $K^4$-building game: two players alternately claim edges of an infinite complete graph. Each player's aim is to claim all six edges on some vertex set of size four for themself. The first player to accomplish this goal is declared the winner of the game. We present a winning strategy which guarantees a win for the first player.

math.CO↗
arXiv2023

Menger's Theorem in bidirected graphs

Bidirected graphs are a generalisation of directed graphs that arises in the study of undirected graphs with perfect matchings. Menger's famous theorem - the minimum size of a set separating two vertex sets $X$ and $Y$ is the same as the maximum number of disjoint paths connecting them - is generally not true in bidirected graphs. We introduce a sufficient condition for $X$ and $Y$ which yields a version of Menger's Theorem in bidirected graphs that in particular implies its directed counterpart.

math.CO↗
arXiv2023

Decomposition of (infinite) digraphs along directed 1-separations

We introduce torsoids, a canonical structure in matching covered graphs, corresponding to the bricks and braces of the graph. This allows a more fine-grained understanding of the structure of finite and infinite directed graphs with respect to their 1-separations.

math.CO↗
arXiv2022

Maker-Breaker games on $ K_{\omega_1}$ and $K_{\omega,\omega_1}$

We investigate Maker-Breaker games on graphs of size $\aleph_1$ in which Maker's goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega,\omega_1}$-game under ZFC+MA+$\neg$CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega_1}$-game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.

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