Flip colouring of graphs
Yair Caro, Josef Lauri, Xandru Mifsud, Raphael Yuster and, Christina Zarb

TL;DR
This paper proves the existence of specific red/blue edge-colored graphs with prescribed degrees and neighborhood properties, extending to multiple colors and larger neighborhoods, with several explicit constructions and related graph classes.
Contribution
It establishes the existence of graphs with fixed red and blue degrees and neighborhood edge counts, extending the theory to multiple colors and larger neighborhoods, with new explicit constructions.
Findings
Existence of graphs with prescribed red/blue degrees and neighborhood properties
Extension to multi-color graphs and larger neighborhoods
Explicit constructions and relationships with known graph classes
Abstract
It is proved that for integers such that , there exists a red/blue edge-colored graph such that the red degree of every vertex is , the blue degree of every vertex is , yet in the closed neighborhood of every vertex there are more blue edges than red edges. The upper bound is best possible for any . We further extend this theorem to more than two colours, and to larger neighbourhoods. A useful result required in some of our proofs, of independent interest, is that for integers such that , there exists an -regular graph in which each open neighborhood induces precisely edges. Several explicit constructions are introduced and relationships with constant linked graphs, -regular graphs and vertex transitive graphs are revealed.
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.
Taxonomy
Topicsgraph theory and CDMA systems · Advanced Graph Theory Research
