On Alternating 6-Cycles in Edge-Coloured Graphs

TL;DR

We study the maximum density of copies of the alternating 6-cycle $C_6^A$ in red/blue edge-coloured graphs and prove $t(C_6^A,G) \le (1/2)^6$, with equality asymptotically achieved by a uniformly random colouring on large cliques. This resolves the first open case of Basit, Granet, Horsley, Kündgen and Staden's semi-inducibility problem. The proof employs Razborov's flag algebra framework, reducing to injective homomorphisms and expressing $t_{inj}(C_6^A,G)$ as a linear combination over the $26$ non-isomorphic colourings of $K_{3,3}$, aided by a positive semidefinite $8 \times 8$ matrix on a collection of flags to derive the bound $t_{inj}(C_6^A,G) \le (1/2)^6 + o(1)$; the standard blow-up argument then yields the bound for $t(C_6^A,G)$. The work highlights the role of quasirandomness in the extremal case and discusses computational simplifications and related independent developments, suggesting potential extensions of the flag-algebra approach to nearby problems.

Abstract

In this short note, we use flag algebras to prove that the number of colour alternating 6-cycles in a red/blue colouring of a large clique is asymptotically maximized by a uniformly random colouring. This settles the first open case of a problem of Basit, Granet, Horsley, Kündgen and Staden.

Figures (2)

• Figure 1: The 26 edge colourings of $K_{3,3}$ up to isomorphism. Red edges are drawn as solid lines and blue edges are drawn as dashed lines.
• Figure 2: The flags $R_1,\dots,R_8$ and $B_1,\dots,B_8$. The roots are depicted with square nodes.

Theorems & Definitions (7)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• proof : Proof of Theorem \ref{['th:main']}, assuming Theorem \ref{['th:injective']}
• Proposition 2.4
• proof
• proof : Proof of Theorem \ref{['th:injective']}