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It follows by Bixbys Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either $mathrm{co}(Mbackslash e)$, the cosimplification of $Mbackslash e$, or $mathrm{si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both $mathrm{co}(Mbackslash e)$ and $mathrm{si}(M/e)$ are $3$-connected. Calling such an element elastic, in this paper we show that if $|E(M)|ge 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans.
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A finite graph $G$ is said to be {em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $gin G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph $G=(V, E)$, either $G_v^{G(v)}$ is $2$--transitive or $G_v^{G(v)}$ is of rank $3$ and $G$ has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types.
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most $48$ elements.
A multigraph is exactly k-edge-connected if there are exactly k edge-disjoint paths between any pair of vertices. We characterize the class of exactly 3-edge-connected graphs, giving a synthesis involving two operations by which every exactly 3-edge-connected multigraph can be generated. Slightly modified syntheses give the planar exactly 3-edge-connected graphs and the exactly 3-edge-connected graphs with the fewest possible edges.
Let $cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $cX$-matroid if each set in $cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $cX$-matroid in the weak order poset of all $cX$-matroids on $E$, and characterizing its rank function when it exists.
We investigate valuated matroids with an additional algebraic structure on their residue matroids. We encode the structure in terms of representability over stringent hyperfields. A hyperfield $H$ is {em stringent} if $aboxplus b$ is a singleton unless $a=-b$, for all $a,bin H$. By a construction of Marc Krasner, each valued field gives rise to a stringent hyperfield. We show that if $H$ is a stringent skew hyperfield, then the vectors of any weak matroid over $H$ are orthogonal to its covectors, and we deduce that weak matroids over $H$ are strong matroids over $H$. Also, we present vector axioms for matroids over stringent skew hyperfields which generalize the vector axioms for oriented matroids and valuated matroids.
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