On Edge Coloring of Multigraphs

Let $\Delta(G)$ and $\chi'(G)$ be the maximum degree and chromatic index of a graph $G$, respectively. Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) made the following conjecture: Every multigraph $G$ satisfies $\chi'(G) \le \max\{ \Delta(G) + 1, \Gamma(G) \}$, where $\Gamma(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\rceil$ is the density of $G$. In this paper, we present a polynomial-time algorithm for coloring any multigraph with $\max\{ \Delta(G) + 1, \Gamma(G) \}$ many colors, confirming the conjecture algorithmically. Since $\chi'(G)\geq \max\{ \Delta(G), \Gamma(G) \}$, this algorithm gives a proper edge coloring that uses at most one more color than the optimum. As determining the chromatic index of an arbitrary graph is $NP$-hard, the $\max\{ \Delta(G) + 1, \Gamma(G) \}$ bound is best possible for efficient proper edge coloring algorithms on general multigraphs, unless $P=NP$

Stanley's Major Contributions to Ehrhart Theory

This expository paper features a few highlights of Richard Stanley's extensive work in Ehrhart theory, the study of integer-point enumeration in rational polyhedra. We include results from the recent literature building on Stanley's work, as well as several open problems.Comment: 9 pages; to appear in the 70th-birthday volume honoring Richard Stanle

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

A note on hierarchical hubbing for a generalization of the VPN problem

Robust network design refers to a class of optimization problems that occur when designing networks to efficiently handle variable demands. The notion of "hierarchical hubbing" was introduced (in the narrow context of a specific robust network design question), by Olver and Shepherd [2010]. Hierarchical hubbing allows for routings with a multiplicity of "hubs" which are connected to the terminals and to each other in a treelike fashion. Recently, Fr\'echette et al. [2013] explored this notion much more generally, focusing on its applicability to an extension of the well-studied hose model that allows for upper bounds on individual point-to-point demands. In this paper, we consider hierarchical hubbing in the context of a previously studied (and extremely natural) generalization of the hose model, and prove that the optimal hierarchical hubbing solution can be found efficiently. This result is relevant to a recently proposed generalization of the "VPN Conjecture".Comment: 14 pages, 1 figur