# Progress on the 123-Conjecture

## Document Type

Presentation Abstract

## Presentation Date

5-7-2014

## Abstract

A *k*-edge-weighting of a graph *G* is a map *w *: E(*G*) ↦ {1,2,...,*k*}. We say that a *k-*edge-weighting induces a proper coloring of *V*(*G*) if for all adjacent vertices *u*, *v *∈ *V*(*G*) the sum of the weights of edges incident to *u* is different from the sum of the weights of edges incident to *v*. In 2004, Karonski, Łuczak, and Thomason conjectured that for any connected graph *G* such that ∣V(*G*)∣ ≥ 3, there exists a 3-edge-weighting that induces a proper coloring of *V*(*G*) . This assertion is known as the 123-Conjecture. In 2010, Kalkowski, Karonski, and Pfender showed that for any such graph *G* there exists a 5-edge-weighting that induces a proper coloring of *V*(*G*).

We confirm the 123-Conjecture for the Kneser graph, the generalized Kneser graph, and any complete *k*-partite graph. Our proofs make use of a technique of alternately weighting collections of edges with 1s and 3s. We also apply the Local Lemma to this problem to show that for 4-regular graphs, there exist 4- edge-weightings that induce proper colorings of the graphs. Additionally, addressing a question of Khatirinejad, Naserasr, Newman, Seamone, and Stevens, we show that for any tree *T* there are at least two non-isomorphic 3- edge-weightings that induce a proper coloring of *V*(*T*).

## Recommended Citation

Fouts, Cody, "Progress on the 123-Conjecture" (2014). *Colloquia of the Department of Mathematical Sciences*. 454.

https://scholarworks.umt.edu/mathcolloquia/454

## Additional Details

Presentation of Master's Project.

Master's Committee: Dr. Cory Palmer, Chair (Mathematical Sciences), Dr. Mark Kayll (Mathematical Sciences), Dr. George McRae (Mathematical Sciences)Wednesday, May 7 at 3:10 p.m. in Math 108