In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.If you have corrections, updates or new results which fit the scope of this work, please contact Colton Magnant at coltonmagnant@yahoo.com.
DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each k ∈ {3, 4, 5, 6}, every planar graph without C k is 4-choosable. Furthermore, Jin, Kang, and Steffen [9] showed that for each k ∈ {3, 4, 5, 6}, every signed planar graph without C k is signed 4-choosable. In this paper, we show that for each k ∈ {3, 4, 5, 6}, every planar graph without C k is 4-DP-colorable, which is an extension of the above results.
Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k‐edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai–Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of k.
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