Semi-Inducibility of some small graphs

TL;DR

This work investigates the density-dependent semi-inducibility problem for small red/blue graphs, introducing $I(H, \beta)$ as the asymptotic maximum of copies of $H$ in a red/blue complete graph with red edge density $\beta$. Through a blend of double counting, degree-sum analysis, and flag-algebras, the authors obtain sharp results for several 4- and 5-vertex graphs and provide general results for trees and stars. Key findings include the exact determination of $I(AP_4, \beta) = $ $\beta^2(1-\beta)$ for all $\beta$, the exact forms for $I(D(s), \beta)$ in a large $\beta$-range, precise bounds and conjectures for $I(AC_4, \beta)$, and a stable description of extremals for $P_{EENN}$ and general trees at high density. The work advances the understanding of semi-inducibility and highlights rich open questions for remaining small graphs and density regimes, with methods spanning combinatorial counting and plane-flag techniques.

Abstract

Let $H$ be a fixed graph whose edges are colored red and blue and let $β\in [0,1]$. Let $I(H, β)$ be the (asymptotically normalized) maximum number of copies of $H$ in a large red/blue edge-colored complete graph $G$, where the density of red edges in $G$ is $β$. This refines the problem of determining the semi-inducibility of $H$, which is itself a generalization of the classical question of determining the inducibility of $H$. The function $I(H, β)$ for $β\in [0,1]$ was not known for any graph $H$ on more than three vertices, except when $H$ is a monochromatic clique (Kruskal-Katona) or a monochromatic star (Reiher-Wagner). We obtain sharp results for some four and five vertex graphs, addressing several recent questions posed by various authors. We also obtain some general results for trees and stars. Many open problems remain.

Figures (7)

• Figure 1: Alternating path $AP_4$ and Alternating 4-cycle $AC_4$.
• Figure 2: The graph $D(5)$.
• Figure 3: Path $P_{EENN}$.
• Figure 4: The upper was bound obtained by flag algebras calculation on 7 vertices and the lower bound obtained by a construction.
• Figure 5: Red and green are bounds from clique and isolated vertices and the complement. Their maximum is $I(P_{EENN},\beta)$. Blue is a regular graph, perhaps it is the minimum.

Theorems & Definitions (31)

• Definition 1
• Definition 2
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1
• Theorem 1.5
• Lemma 2.1: Nikiforov Nikiforov2005
• proof