Perfect Matchings in Claw-free Cubic Graphs

Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure

On factors of 4-connected claw-free graphs

We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is Hamiltonian, i.e. has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass-free graphs, i.e. graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjecture 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths. \u

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Subalgebras of the Fomin-Kirillov algebra

The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, we define $\mathcal E_G$ to be the subalgebra of $\mathcal E_n$ generated by the edges of $G$. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $\mathcal E_G$ is a free $\mathcal E_H$-module for any $H\subseteq G$, and if $\mathcal E_G$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $\mathcal E_G$ when $G$ is a simply-laced finite Dynkin diagram or a cycle, in particular showing that $\mathcal E_G$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $\mathcal E_{\tilde{D}_n}$, $\mathcal E_{\tilde{E}_6}$, and $\mathcal E_{\tilde{E}_7}$, as well as for which graphs $G$ on six vertices $\mathcal E_G$ is finite-dimensional.Comment: 38 pages, 10 figures + appendi