Flip colouring of graphs

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 $b, r$ such that $3 \leq b < r \leq \binom{b+1}{2} - 1$, there exists a red/blue edge-colored graph such that the red degree of every vertex is $r$, the blue degree of every vertex is $b$, yet in the closed neighborhood of every vertex there are more blue edges than red edges. The upper bound $r \le \binom{b+1}{2}-1$ is best possible for any $b \ge 3$. 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 $r,t$ such that $0 \leq t \le \frac{r^2}{2} - 5r^{3/2}$, there exists an $r$-regular graph in which each open neighborhood induces precisely $t$ edges. Several explicit constructions are introduced and relationships with constant linked graphs, $(r,b)$-regular graphs and vertex transitive graphs are revealed.