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).

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

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