#### Title of Abstract

F-WORM Graph Colorings: Forbidding Trees, Paths, and Stars

#### Session Title

Poster Session 1

#### Faculty Mentor

Arran Hamm, Ph. D.| Jessie Hamm, Ph. D.

#### College

College of Arts and Sciences

#### Department

Mathematics

#### Abstract

Given graphs G and F, define an F-WORM coloring of G to be an assignment of colors to the vertices of G so that there is no “rainbow” (meaning all vertices are different colors) or “monochromatic” (meaning all vertices are the same color) copy of F. For this type of problem, G is usually chosen so that it contains many copies of F and in this context we regard G as the “host” graph and F as the “target” graph. This type of graph coloring problem was introduced in 2015 by Goddard, Wash, and Xu as something of a blend of Ramsey Theory and Anti-Ramsey Theory which seek to avoid monochromatic and rainbow structures, respectively. Since the introduction of WORM coloring, there have been only been a few publications on this topic. In an effort to expand the body of work in this research area, we examined several WORM coloring problems with various host graph/target graph pairs. In particular, we obtained results for F-WORM coloring when F is a tree, a path, and a star within complete bipartite graphs, cycles, and hypercubes, respectively.

#### Previously Presented/Performed?

Winthrop University Showcase of Undergraduate Research and Creative Endeavors, Rock Hill, SC, April 2023.

#### Type of Presentation

Poster presentation

#### Grant Support?

Supported by an SC-INBRE grant from the National Institute for General Medical Sciences (P20GM103499).

#### Start Date

15-4-2023 12:00 PM

F-WORM Graph Colorings: Forbidding Trees, Paths, and Stars

Given graphs G and F, define an F-WORM coloring of G to be an assignment of colors to the vertices of G so that there is no “rainbow” (meaning all vertices are different colors) or “monochromatic” (meaning all vertices are the same color) copy of F. For this type of problem, G is usually chosen so that it contains many copies of F and in this context we regard G as the “host” graph and F as the “target” graph. This type of graph coloring problem was introduced in 2015 by Goddard, Wash, and Xu as something of a blend of Ramsey Theory and Anti-Ramsey Theory which seek to avoid monochromatic and rainbow structures, respectively. Since the introduction of WORM coloring, there have been only been a few publications on this topic. In an effort to expand the body of work in this research area, we examined several WORM coloring problems with various host graph/target graph pairs. In particular, we obtained results for F-WORM coloring when F is a tree, a path, and a star within complete bipartite graphs, cycles, and hypercubes, respectively.