Flip colouring of graphs II

TL;DR

This paper advances the theory of flip colourings by (i) producing small $(b,r)$-flip graphs for $4\le b<r<b+2\left\lfloor\frac{b+2}{6}\right\rfloor^2$ via Cayley-graph constructions rooted in sum-free subsets, yielding a new upper bound on $h(b,r)$, and (ii) establishing both upper and lower bounds on $q(k)$, the extent of fixed early terms in $k$-flip sequences, including a lower bound showing $q(k) \gtrsim k/4$ and a general upper bound $q(k) < k/3$ (in specific residue classes) or $< \lceil k/2\rceil$ otherwise. The methods synthesize graph products, packing, and Abelian-group Cayley graphs with sum-free sets to connect local neighbourhood constraints to global realizability, enabling controlled constructions and unbounded behaviour in large final colours. The results illuminate the growth of $h$ and $q(k)$, reveal non-polynomial growth in some regimes, and pose open problems regarding polynomial bounds tied to the index $p(k)$ and the feasible parameter ranges for $h(b,r)$. The work integrates combinatorial design with algebraic graph theory to push the boundary of what is constructible under flip-colouring constraints.

Abstract

We give results concerning two problems on the recently introduced \textit{flip colourings of graphs}. For positive integers $b, r$ with $b < r$, we say that a $b + r$ regular graph is a $(b,r)$-\textit{flip graph} if there exists a red/blue edge colouring such that the red degree of every vertex is $r$, the blue degree of every vertex is $b$, yet in the closed neighbourhood of every vertex there are more blue edges than red edges. We prove that for integers $b, r$ with $4 \leq b < r < b + 2 \left\lfloor\frac{b+2}{6}\right\rfloor^2$, small constructions of $(b,r)$-flip graphs on $Θ(b+r)$ vertices are possible. Furthermore, we prove that there exist $k$-flip sequences $(a_1, \dots, a_k)$ where $k > 4$, such that $a_k$ can be arbitrarily large whilst $a_i$ is constant for $1 \leq i < \frac{k}{4}$.

Figures (6)

• Figure 1: Smallest known $(3, 4)$-flip graph, with the subgraph induced by the closed neighbourhood of any vertex $v$ illustrated on the right. This is a $(3, 4)$-flip graph since: $\deg_{\mathsf{blue}}(v) = 3 < 4 = \deg_{\mathsf{red}}(v)$ but $e_{\mathsf{blue}}[v] = 7 > 6 = e_{\mathsf{red}}(v)$.
• Figure 2: Illustration of Lemma \ref{['SPColourLemma']}, with the closed neighbourhood of $w = (u, v)$ in $K_3 \boxtimes P_3$ highlighted.
• Figure 3: Illustration of Lemma \ref{['CPColourLemma']}, with the closed neighbourhood of $w = (u, v)$ in $K_3 \ \square \ P_3$ highlighted.
• Figure 4: Illustration of the counting argument in the proof of Proposition \ref{['cayley_union_counting']}, where the red edge $\{u, x\}$ between two blue neighbours of $1_\Gamma$ corresponds to two blue edges, each incident to a blue and red neighbour of $1_\Gamma$.
• Figure 5: Illustration of the closed neighbourhood of $1$ in the Cayley graph construction for $(b, r) = (6, 7)$ and $n = 56$ in the proof of Theorem \ref{['newBound']}, with the choice of $R_0$, $T_0$ and $T_2$ highlighted.

Theorems & Definitions (23)

• Theorem 1.1: Theorem 3.1, caro2023flip
• Theorem 1.2: Corollary 3.6, caro2023flip
• Theorem 1.3: Theorem 4.1, caro2023flip
• Theorem 1.4: Theorem 4.3, caro2023flip
• Definition 2.1: Strong product
• Lemma 2.1
• Definition 2.2: Cartesian product
• Lemma 2.2
• Definition 2.3: Packing
• Proposition 2.3