arXiv subjects

Matt DeVos

Publications and source records attributed to Matt DeVos.

At least 37 records · Page 2Linked to original sources

arXiv2018

The structure of graphs with no K_{3,3} immersion

The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either $K_{3,3}$ or $K_5$ as a minor. Using this Wagner obtained a precise description of all graphs with no $K_{3,3}$ minor and all graphs with no $K_5$ minor. Similar results have been achieved for the class of graphs with no $H$-minor for a number of small graphs $H$. In this paper we give a precise structure theorem for graphs which do not contain $K_{3,3}$ as an immersion. This strengthens an earlier theorem of Giannopoulou, Kamiński, and Thilikos that gives a rough description of the class of graphs with no $K_{3,3}$ or $K_5$ immersion.

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arXiv2018

The structure of graphs with no W_4 immersion

This paper gives a precise structure theorem for the class of graphs which do not contain $W_4$ as an immersion. This strengthens a previous result of Belmonte at al. that gives a rough description of this class. In fact, we prove a stronger theorem concerning rooted immersions of $W_4$ where one terminal is specified in advance. This stronger result is key in a forthcoming structure theorem for graphs with no $K_{3,3}$ immersion.

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arXiv2018

Non-monochromatic Triangles in a 2-Edge-Coloured Graph

Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$. We conjecture a natural generalization to partitions with more blocks.

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arXiv2017

Almost balanced biased graph representations of frame matroids

Given a 3-connected biased graph $Ω$ with a balancing vertex, and with frame matroid $F(Ω)$ nongraphic and 3-connected, we determine all biased graphs $Ω'$ with $F(Ω') = F(Ω)$. As a consequence, we show that if $M$ is a 4-connected nongraphic frame matroid represented by a biased graph $Ω$ having a balancing vertex, then $Ω$ essentially uniquely represents $M$. More precisely, all biased graphs representing $M$ are obtained from $Ω$ by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.

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arXiv2017

Whitney's Theorem for 2-Regular Planar Digraphs

A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embedding of the underlying graph in a closed surface with the added property that for every vertex~$v$, the two edges directed away from $v$ are not consecutive in the local rotation around $v$. In other words, at each vertex the incident edges are oriented in-out-in-out. The goal of this article is to provide an analogue of Whitney's theorem on planar embeddings in the setting of 2-regular digraphs. In the course of doing so, we note that Tutte's Theorem on peripheral cycles also has a natural analogue in this setting.

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arXiv2017

A note on counting flows in signed graphs

Tutte initiated the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $Γ$ of order $n$, the number of nowhere-zero $Γ$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a finite group $Γ$, let $ε_2(Γ)$ be the largest integer $d$ so that $Γ$ has a subgroup isomorphic to $\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the number of nowhere-zero $Γ$-flows in $G$ for every abelian group $Γ$ with $ε_2(Γ) = d$ and $|Γ| = 2^d n$. Beck and Zaslavsky had previously established the special case of this result when $d=0$ (i.e., when $Γ$ has odd order).

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arXiv2016

On excluded minors of connectivity 2 for the class of frame matroids

We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $\mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N \in \mathcal{E}$ or $N$ is a 2-sum of $U_{2,4}$ and a 3-connected non-binary frame matroid.

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arXiv2015

A new proof of Seymour's 6-flow theorem

Tutte's famous 5-flow conjecture asserts that every bridgeless graph has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. Here we give (two versions of) a new proof of Seymour's Theorem. Both are roughly equal to Seymour's in terms of complexity, but they offer an alternative perspective which we hope will be of value.

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arXiv2015

Period Preserving Properties of an Invariant from the Permanent of Signed Incidence Matrices

A 4-point Feynman diagram in scalar $ϕ^4$ theory is represented by a graph $G$ which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of $G$ which is invariant under a number of meaningful graph operations - namely, planar duality, the Schnetz twist, and it also does not depend on the choice of vertex which was deleted to form $G$. In this article we study a graph invariant we call the graph permanent, which was implicitly introduced in a paper by Alon, Linial and Meshulam. The graph permanent applies to any graph $G = (V,E)$ for which $|E|$ is a multiple of $|V| - 1$ (so in particular to graphs obtained from a 4-regular graph by removing a vertex). We prove that the graph permanent, like the period, is invariant under planar duality and the Schnetz twist when these are valid operations, and we show that when $G$ is obtained from a $2k$-regular graph by deleting a vertex, the graph permanent does not depend on the choice of deleted vertex.

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arXiv2015

The 4-Regular Edge-Transitive Graphs of Girth 4

This paper presents a characterization of edge-transitive graphs which are four regular and have girth four. This class consists of four infinite families plus four exceptional graphs.

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arXiv2015

Packing Steiner Trees

Let $T$ be a distinguished subset of vertices in a graph $G$. A $T$-\emph{Steiner tree} is a subgraph of $G$ that is a tree and that spans $T$. Kriesell conjectured that $G$ contains $k$ pairwise edge-disjoint $T$-Steiner trees provided that every edge-cut of $G$ that separates $T$ has size $\ge 2k$. When $T=V(G)$ a $T$-Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved that Kriesell's conjecture holds when $2k$ is replaced by $24k$, and recently West and Wu have lowered this value to $6.5k$. Our main result makes a further improvement to $5k+4$.

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arXiv2015

Packing triangles in weighted graphs

Tuza conjectured that for every graph $G$, the maximum size $ν$ of a set of edge-disjoint triangles and minimum size $τ$ of a set of edges meeting all triangles satisfy $τ\leq 2ν$. We consider an edge-weighted version of this conjecture, which amounts to packing and covering triangles in multigraphs. Several known results about the original problem are shown to be true in this context, and some are improved. In particular, we answer a question of Krivelevich who proved that $τ\leq 2ν^*$ (where $ν^*$ is the fractional version of $ν$), and asked if this is tight. We prove that $τ\leq 2ν^*-\frac{1}{\sqrt{6}}\sqrt{ν^*}$ and show that this bound is essentially best possible.

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arXiv2014

Forbidden minors for graphs with no first obstruction to parametric Feynman integration

We give a characterization of 3-connected graphs which are planar and forbid cube, octahedron, and $H$ minors, where $H$ is the graph which is one $Δ-Y$ away from each of the cube and the octahedron. Next we say a graph is Feynman 5-split if no choice of edge ordering gives an obstruction to parametric Feynman integration at the fifth step. The 3-connected Feynman 5-split graphs turn out to be precisely those characterized above. Finally we derive the full list of forbidden minors for Feynman 5-split graphs of any connectivity.

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arXiv2014

When does a biased graph come from a group labelling?

A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on $G$ arise from orienting $G$ and then labelling the edges of $G$ with elements of a group $Γ$. In this case, we may define a biased graph by declaring a cycle to be balanced if the product of the labels on its edges is the identity, with the convention that we take the inverse value for an edge traversed backwards. Our first result gives a natural topological characterisation of biased graphs arising from group-labellings. In the second part of this article, we use this theorem to construct some exceptional biased graphs. Notably, we prove that for every $m \ge 3$ and $\ell$ there exists a minor minimal not group labellable biased graph on $m$ vertices where every pair of vertices is joined by at least $\ell$ edges. Finally, we show that these results extend to give infinite families of excluded minors for certain families of frame and lift matroids.

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arXiv2014

Graphical representations of graphic frame matroids

A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a biased graph; this class includes graphic matroids, bicircular matroids, and Dowling geometries. Whitney characterized which graphs have isomorphic cycle matroids, and Matthews characterised which graphs have isomorphic graphic bicircular matroids. In this paper, we give a characterization of which biased graphs give rise to isomorphic graphic frame matroids.

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arXiv2013

Flows on Bidirected Graphs

The study of nowhere-zero flows began with a key observation of Tutte that in planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of k-tensions). Tutte conjectured that every graph without a cut-edge has a nowhere-zero 5-flow. Seymour proved that every such graph has a nowhere-zero 6-flow. For a graph embedded in an orientable surface of higher genus, flows are not dual to colourings, but to local-tensions. By Seymour's theorem, every graph on an orientable surface without the obvious obstruction has a nowhere-zero 6-local-tension. Bouchet conjectured that the same should hold true on non-orientable surfaces. Equivalently, Bouchet conjectured that every bidirected graph with a nowhere-zero $\mathbb{Z}$-flow has a nowhere-zero 6-flow. Our main result establishes that every such graph has a nowhere-zero 12-flow.

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arXiv2013

Highly arc-transitive digraphs -- counterexamples and structure

We resolve two problems of [Cameron, Praeger, and Wormald -- Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each `building block' is a finite bipartite graph that is not a disjoint union of complete bipartite graphs. This was conjectured impossible in the above paper. We also describe the structure of 2-ended highly arc transitive digraphs in more generality, although complete characterization remains elusive.

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arXiv2013

The Structure of Critical Product Sets

Let $G$ be a multiplicative group, let $A,B \subseteq G$ be finite and nonempty, and define the product set $AB = {ab \mid $a \in A$ and $b \in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on $|AB|$, and then to determine structural properties of $A$ and $B$ under the assumption that $|AB|$ is small. We focus on the extreme case when $|AB| < |A| + |B|$, and call any such pair $(A,B)$ \emph{critical}. In the case when $|G|$ is prime, the Cauchy-Davenport Theorem asserts that $|AB| \ge \min {|G|, |A| + |B| - 1}$, and Vosper refined this result by classifying all critical pairs in these groups. For abelian groups, Kneser proved a natural generalization of Cauchy-Davenport by showing that there exists $H \le G$ so that $|AB| \ge |A| + |B| - |H|$ and $ABH = AB$. Kemperman then proved a result which characterizes the structure of all critical pairs in abelian groups. Our main result gives a classification of all critical pairs in an arbitrary group $G$. As a consequence of this we derive the following generalization of Kneser's Theorem to arbitrary groups: There exists $H \le G$ so that $|AB| \ge |A| + |B| - |H|$ and so that for every $y \in AB$ there exists $x \in G$ so that $y(x^{-1} H x) \subseteq AB$.

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