Almost-rainbow edge-colorings of some small subgraphs
Elliot Krop, Irina Krop

TL;DR
This paper improves bounds on the minimum number of colors needed to edge-color complete graphs and bipartite graphs so that small subgraphs have a certain diversity of colors, advancing understanding in graph coloring problems.
Contribution
The paper provides new bounds and constructions for edge-colorings of complete and bipartite graphs ensuring diverse coloring of small subgraphs, improving previous results.
Findings
Improved bounds on f(n,p,q) for complete graphs.
Constructed n-color edge-coloring of K_{n,n} with 3 colors on every C_4.
Enhanced previous bounds by Axenovich, Erdős, and Gyárfás.
Abstract
Let be the minimum number of colors necessary to color the edges of so that every is at least -colored. We improve current bounds on the {7/4}n-3{5/6}n+1\leq f(n,4,5)n\not\equiv 1 \pmod 3f(n,4,5)\leq n-1G=K_{n,n}GC_4\subseteq G$ is colored by at least three colors. This improves the best known upper bound of M. Axenovich, Z. F\"uredi, and D. Mubayi.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph Labeling and Dimension Problems · graph theory and CDMA systems
