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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.
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Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.
Motivated by generalizing Khovanovs categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,mathcal{A},F)$. We find that CW posets, that is, face posets of regular CW complexes, satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category of tuples $(P,mathcal{A},F,c)$, where $c$ is a certain ${1,-1}$-coloring of the cover relations in $P$, and show the cohomology arising from a tuple $(P,mathcal{A},F,c)$ is functorial, and independent of the coloring $c$ up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants: anything expressible as a rank-alternating sum over a thin poset.
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.
We consider the problem of determining the maximum order of an induced vertex-disjoint union of cliques in a graph. More specifically, given some family of graphs $mathcal{G}$ of equal order, we are interested in the parameter $a(mathcal{G}) = min_{G in mathcal{G}} max { |U| : U subseteq V, G[U] text{ is a vertex-disjoint union of cliques} }$. We determine the value of this parameter precisely when $mathcal{G}$ is the family of comparability graphs of $n$-element posets with acyclic cover graph. In particular, we show that $a(mathcal{G}) = (n+o(n))/log_2 (n)$ in this class.
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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