Multiply Balanced Edge Colorings of Multigraphs
Amin Bahmanian, Chris Rodger

TL;DR
This paper proves a general theorem on balanced edge colorings of multigraphs, extending existing results and addressing complex amalgamation and disentangling problems with applications to Hamilton decompositions.
Contribution
It introduces a broad theorem that generalizes previous amalgamation results, allowing for more complex graph structures and addressing multiple notions of balance.
Findings
Generalized amalgamation theorem for multigraphs
Corollary on Hamilton decompositions of specific graph families
Addresses connectivity and fairness in edge colorings
Abstract
In this paper, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out among the vertices in ways that are fair with respect to several notions of balance (such as between pairs of vertices, degrees of vertices in the both graph and in each color class, etc). The connectivity of color classes is also addressed. Most results in the literature on amalgamations focus on the disentangling of amalgamated complete graphs and complete multipartite graphs. Many such results follow as immediate corollaries to the main result in this paper, which addresses amalgamations of graphs in general, allowing for example the final graph to have multiple edges. A new corollary of the main theorem is the settling of the existence of…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
