Graph Labelings and Structures, Sumsets, and the Connections Between Them

Author

• Runze Wang

Date

2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

David Grynkiewicz

Committee Member

Alexander Gavrilyuk

Committee Member

Ben McCarty

Committee Member

David Grynkiewicz

Committee Member

Nathan Lindzey

Abstract

The study of graph labelings and structures is a vital facet of graph theory, while the study of sumsets plays a central role in additive combinatorics. There are four chapters in this dissertation: Chapter 1 is an introduction, Chapter 2 is about graph labelings and structures, Chapter 3 is about sumsets, and Chapter 4 is about the connections between graph labelings and sumsets. In Chapter 2, we define the \emph{$(m,1)$-edge coloring game}, alternatively played by Maker and Breaker on a graph with a set of colors. Maker colors $m$ edges on each turn, but Breaker only colors one edge on each turn. They make sure that adjacent edges get distinct colors. Maker wins if eventually every edge is colored; Breaker wins if at some point, the player who is playing cannot color any edge. In this chapter, we study the $(m,1)$-edge coloring game on trees, caterpillars, and wheels. In Chapter 3, we prove a Kneser-type result for multi-representable sumsets. For two sets $A$ and $B$ in an abelian group $G$, the \emph{$t$-representable sumset} of $A$ and $B$, denoted by $A+_t B$, is the set of elements in $G$ each with at least $t$ representations of the form $a+b$, where $a\in A$ and $b\in B$. The main result in this chapter generalizes Pollard's theorem to general abelian groups; provides strong structural information when $\sum_{i=1}^t |A+_i B|$ is small; and improves the main quadratic term in the previous best bound. Chapter 4 is devoted to a graph labeling problem with a background rooted in additive combinatorics. In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this chapter, we explain the connections between the sum index and results in additive combinatorics; determine the sum indices of some graphs; and study the maximum number of edges in a graph with a fixed sum index.

Comments

Data is provided by the student.”

Notes

Open Access

Recommended Citation

Wang, Runze, "Graph Labelings and Structures, Sumsets, and the Connections Between Them" (2026). Electronic Theses and Dissertations Archive. 3943.
https://digitalcommons.memphis.edu/etd/3943