arXiv subjects

Mitchell Minyard

Publications and source records attributed to Mitchell Minyard.

6 recordsLinked to original sources

arXiv2026

Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux

Young tableaux are fundamental objects in algebraic combinatorics and representation theory, with operations such as promotion and jeu de taquin playing a central role in their structure and applications. While these operations are well understood for finite tableaux, their behavior on infinite tableaux has so far been studied mainly within probabilistic frameworks. In this paper, we investigate jeu de taquin on infinite standard Young tableaux from a purely combinatorial and dynamical point of view. We analyze the action of jeu de taquin on infinite shapes, describe the structure of inverse images, and classify tableaux exhibiting periodic, pre-periodic, and recurrent behavior. We also introduce a natural metric on the space of infinite tableaux and show that jeu de taquin defines a chaotic dynamical system in the sense of Devaney. These results extend classical tableau theory to infinite settings and identify connections between combinatorial dynamics and infinite representation-theoretic structures.

math.CO↗

arXiv2025

Chromatic numbers with open and nonzero local modular constraints

In this paper, we explore chromatic numbers subject to various local modular constraints. For fixed $n$, we consider proper integer colorings of a graph $G$ for which the closed and open neighborhood sums have nonzero remainders modulo $n$ and provide bounds for the associated chromatic numbers $\chi_n(G)$ and $\chi_{(n)}(G)$, respectively. In addition, we provide bounds for $\chi_{(n,k)}(G)$, the minimal order of a proper integer coloring of $G$ with open neighborhood sums congruent to $k\mod n$ (when such a coloring exists) as well as precise values for certain families of graphs.

math.CO↗

arXiv2025

Chromatic numbers with closed local modular constraints

Generalizing the notion of odd-sum colorings, a $\mathbb{Z}$-labeling of a graph $G$ is called a closed coloring with remainder $k\mod n$ if the closed neighborhood label sum of each vertex is congruent to $k\mod n$. If such colorings exist, we write $\chi_{n,k}(G)$ for the minimum number of colors used for a closed coloring with remainder $k\mod n$ such that no neighboring vertices have the same color. General estimates for $\chi_{n,k}(G)$ are given along with evaluations of $\chi_{n,k}(G)$ for some finite and infinite order graphs.

math.CO↗

arXiv2024

Robinson-Schensted shapes arising from cycle decompositions

In the symmetric group $S_n$, each element $\sigma$ has an associated cycle type $\alpha$, a partition of $n$ that identifies the conjugacy class of $\sigma$. The Robinson-Schensted (RS) correspondence links each $\sigma$ to another partition $\lambda$ of $n$, representing the shape of the pair of Young tableaux produced by applying the RS row-insertion algorithm to $\sigma$. Surprisingly, the relationship between these two partitions, namely the cycle type $\alpha$ and the RS shape $\lambda$, has only recently become a subject of study. In this work, we explicitly describe the set of RS shapes $\lambda$ that can arise from elements of each cycle type $\alpha$ in cases where $\alpha$ consists of two cycles. To do this, we introduce the notion of an $\alpha$-coloring, where one colors the entries in a certain tableau of shape $\lambda$, in such a way as to construct a permutation $\sigma$ with cycle type $\alpha$ and RS shape $\lambda$.

math.CO↗

arXiv2024

Neighborhood Balanced 3-Coloring

A graph is said to be neighborhood 3-balanced if there exists a vertex labeling with three colors so that each vertex has an equal number of neighbors of each color. We give order constraints on 3-balanced graphs, determine which generalized Petersen and Pappus graphs are 3-balanced, discuss when being 3-balanced is preserved under various graph constructions, give two general characterizations of cubic 3-balanced graphs, and classify cubic 3-balanced graphs of small order.

math.CO↗

arXiv2024

Limits and Periodicity of Metamour $2$-Distance Graphs

Given a finite simple graph $G$, let $\operatorname{M}(G)$ denote its 2-distance graph, in which two vertices are adjacent if and only if they have distance 2 in $G$. In this paper, we consider the periodic behavior of the sequence $G, \operatorname{M}(G), \operatorname{M}^2(G), \operatorname{M}^3(G), \ldots$ obtained by iterating the 2-distance operation. In particular, we classify the connected graphs with period 3, and we partially characterize those with period 2. We then study two families of graphs whose 2-distance sequence is eventually periodic: namely, generalized Petersen graphs and complete $m$-ary trees. For each family, we show that the eventual period is 2, and we determine the pre-period and the two limit graphs of the sequence.

math.CO↗