Parabolic subgroups in FC-type Artin groups

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results of Cumplido, Gebhardt, Gonzales-Meneses and Wiest, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC-type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed by Cumplido et al. using parabolic subgroups. We extend the construction of this complex, called the complex of parabolic subgroups, to FC-type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ-hyperbolic.     

On distance-regular Cayley graphs of generalized dicyclic groups

Let G be a generalized dicyclic group with identity 1. An inverse closed subset S of $G?\{1\}$ is called minimal if $〈S〉=G$ and there exists some $s\in S$ such that $〈S?\{s,s^{?1}\}〉\ne G$. In this paper, we characterize distance-regular Cayley graphs $\mathrm{Cay}(G,S)$ of G under the condition that S is minimal.     

Sufficient Conditions for a Digraph to be Supereulerian

A (di)graph is supereulerian if it contains a spanning eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we analyze a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the first author: if the arc‐connectivity of a digraph is not smaller than its independence number, then the digraph is supereulerian. As a support for this conjecture we prove it for digraphs that are semicomplete multipartite or quasitransitive and verify the analogous statement for undirected graphs.     

Vertex-disjoint rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph of order $n$. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if ${\delta }^c\left(G\right)\ge (n+1)∕2$, then $G$ contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if $n\ge 20$ and ${\delta }^c\left(G\right)\ge (n+2)∕2$, then $G$ contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if ${\delta }^c\left(G\right)\ge (n+k)∕2$, then $G$ contains $k$ vertex-disjoint rainbow triangles. For any integer $k\ge 2$, we show that if $n\ge 16k-12$ and ${\delta }^c\left(G\right)\ge n∕2+k-1$, then $G$ contains $k$ vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of $k$ edge-disjoint rainbow triangles.