Nick Brettell

activity

20152026

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6 papers · 1 filter

math.CO2026

The excluded minors for $Z_{3}$ -gainable and regular biased graphs

Nick Brettell, Rutger Campbell, Daryl Funk +1

We prove that a biased graph is gainable over the group $Z_{3}$ if and only if it contains no minor isomorphic to $(4 K_{2}, \emptyset)$ , $\pm K_{3}$ , or $- K_{4}$ . We develo…

math.CO2026

A Brooks-type theorem for the k-choosability of graphs with maximum local edge-connectivity k

Sam Bastida, Nick Brettell

For a graph $G$ with at least two vertices, the maximum local edge-connectivity of $G$ is the maximum number of edge-disjoint $(u, v)$ -paths over all distinct pairs of vertices $(u,…

math.CO2025

The excluded minors of the class of spike minors

Sam Bastida, Nick Brettell, Rutger Campbell +4

Mayhew et al.\ (2021) posed the problem of showing that the minor-closed class of spikes and their minors has a finite set of excluded minors and describing all of them. In this pa…

math.CO2024

Detachable pairs in $3$ -connected matroids and simple $3$ -connected graphs

Nick Brettell, Charles Semple, Gerry Toft

Let $M$ be a $3$ -connected matroid. A pair ${e, f}$ in $M$ is detachable if $M \ e \ f$ or $M / e / f$ is $3$ -connected. Williams (2015) proved that if $M$ has a…

math.CO2018

$N$ -detachable pairs in 3-connected matroids II: life in $X$

Nick Brettell, Geoff Whittle, Alan Williams

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$ . A pair ${x_{1}, x_{2}} \subseteq E (M)$ is $N$ -detachable if one of the matroids $M / x_{1} / x_{2}$ or $M \backsla…

math.CO2018

$N$ -detachable pairs in 3-connected matroids I: unveiling $X$

Nick Brettell, Geoff Whittle, Alan Williams

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$ . We say that a pair ${x_{1}, x_{2}} \subseteq E (M)$ is $N$ -detachable if one of the matroids $M / x_{1} / x_{2}$ or…

The excluded minors for Z3​-gainable and regular biased graphs

A Brooks-type theorem for the k-choosability of graphs with maximum local edge-connectivity k

The excluded minors of the class of spike minors

Detachable pairs in 3-connected matroids and simple 3-connected graphs

N-detachable pairs in 3-connected matroids II: life in X

N-detachable pairs in 3-connected matroids I: unveiling X

The excluded minors for $Z_{3}$ -gainable and regular biased graphs

Detachable pairs in $3$ -connected matroids and simple $3$ -connected graphs

$N$ -detachable pairs in 3-connected matroids II: life in $X$

$N$ -detachable pairs in 3-connected matroids I: unveiling $X$