Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the n-dimensional hypercube so only two different in-degrees a and b occur. We show that this can be done, for two specified in-degrees, if and only if obvious necessary conditions hold. Namely, we need 0?a,b?n and also there exist non-negative integers s and t so that s+t=n² and as+bt=n²n−1. This is connected to a question arising from constructing a strategy for a “hat puzzle”.     

LLT polynomials,chromatic quasisymmetric functions and graphs with cycles

We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their $\mathrm{e}$-positivity.The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the non-circular combinatorics, we prove several statements regarding the $\mathrm{e}$-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the $\mathrm{e}$-coefficients for the line graph and the cycle graph for both families. We believe that certain $\mathrm{e}$-positivity conjectures hold in all these families above.Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall–Littlewood polynomials.     

The relationship between mathematicians’ pedagogical goals,orientations, and common teaching practices in advanced mathematics

Despite mathematics educators’ research into more effective modes of teaching, lecture is still the dominant mode of instruction in undergraduate mathematics courses. Surveys suggest this is because most mathematicians believe this is the best way to teach. This paper answers a call by mathematics education researchers to explore mathematicians’ needs and goals concerning teaching. We interviewed eight mathematicians about findings in the mathematics education research literature concerning common pedagogical practices of instructors of advanced mathematics classes: “chalk talk,” the presentation of formal and informal content, and teacher questioning. We then analyzed the responses for resources, orientations, and goals that might influence the participants to engage in these practices. We describe how participants believed common lecturing practices allowed them to achieve their goals and aligned with their orientations. We discuss these findings in depth and consider what implications they may have for researchers that aim to change mathematicians’ teaching practices.     

Decomposing a graph into bistars

Bárat and the present author conjectured that, for each tree T  , there exists a natural number _kT$k_T$ such that the following holds: If G   is a _kT$k_T$-edge-connected graph such that |E(T)|$|E(T)|$ divides |E(G)|$|E(G)|$, then G has a T-decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T  . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S(k,k+1)$S(k,k+1)$.     

Competition-reachability of a graph

The reachability r(D) of a directed graph D is the number of ordered pairs of distinct vertices (x,y) with a directed path from x to y. Consider a game associated with a graph G=(V,E) involving two players (maximizer and minimizer) who alternately select edges and orient them. The maximizer attempts to maximize the reachability, while the minimizer attempts to minimize the reachability, of the resulting digraph. If both players play optimally, then the reachability is fixed. Parameters that assign a value to each graph in this manner are called competitive parameters. We determine the competitive-reachability for special classes of graphs and discuss which graphs achieve the minimum and maximum possible values of competitive-reachability.