Chromatic numbers with closed local modular constraints
Daniel Herden, Jonathan Meddaugh, Mark R. Sepanski, William Clark, Adam Kraus, Ellie Matter, Kyle Rosengartner, Elyssa Stephens, John Stephens, Mitchell Minyard, Kingsley Michael, Maricela Ramirez
TL;DR
This paper introduces closed colorings with remainder, defining the modular closed chromatic number chi_{n,k}(G) and connecting it to odd-sum coloring via chi_{2,1}(G) = chi_os(G). It develops general bounds, invariances, and existence criteria, and computes exact values for a wide range of graph families including complete graphs, stars, friendship graphs, paths, complete bipartite graphs, cycles, regular graphs, infinite plane tilings R_3, R_4, R_6, and several trees, as well as generalized Petersen graphs. Key insights include the inequality chi(G) ≤ chi_{n,k}(G) ≤ chi(G) + 1 under the presence of an independent efficient dominating set (IEDS), and gcd-based existence conditions such as (ij−1, n) | (i−1)k for K_{i,j} and parity/gcd criteria for cycles and tilings; phenomena in trees (e.g., caterpillars and rooted binary trees) reveal chaotic behavior and rich structure. The results broaden the theory of modular graph colorings, provide constructive techniques (including level-constant labelings and power-series methods) and open questions, especially around the exact classifications for generalized Petersen graphs and certain high-divisibility cases.
Abstract
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 $χ_{n,k}(G)$ 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 $χ_{n,k}(G)$ are given along with evaluations of $χ_{n,k}(G)$ for some finite and infinite order graphs.