Edge version of the inducibility via the entropy method

TL;DR

Addresses the edge-inducibility problem by studying $ρ(H,m)$, the maximum number of induced copies of $H$ in graphs with $m$ edges, and proves $ρ(H,m) = Θ(m^{α_f(H)})$ where $α_f(H)$ is the fractional independence number. The authors employ the entropy method, together with Shearer's lemma, to derive sharp upper bounds for induced paths and cycles, and construct blow-up graphs to match lower bounds, yielding near-tight results for $P_k$ and $C_k$ and exact constants for certain small cycles. They conjecture the asymptotic $ρ(C_k,m) = (1+o(1))(m/k)^{k/2}$ for all $k≥5$, and provide tight or near-tight bounds across the cycle spectrum, including a special treatment of $C_6$. Overall, the work furnishes a unified entropy-based framework for edge-inducibility under edge-count constraints and clarifies the role of the fractional independence number in these extremal problems.

Abstract

The inducibility of a graph $H$ is about the maximum number of induced copies of $H$ in a graph on $n$ vertices. We consider its edge version, that is, the maximum number of induced copies of $H$ in a graph with $m$ edges. Let $c(G,H)$ be the number of induced copies of $H$ in $G$ and $ρ(H,m) = \max \{c(G,H) \mid |E(G)| = m\}$. For any graph $H$, we prove that $ρ(H,m) = Θ(m^{α_f(H)})$ where $α_f(H)$ is the fractional independence number of $H$. Therefore, we now focus on the constant factor in front of $m^{α_f(H)}$. In this paper, we give some results of $ρ(H,m)$ when $H$ is a cycle or path. We conjecture that for any cycle $C_k$ with $k \ge 5$, $ρ(C_k,m)= (1+o(1))\left( m/k\right)^{k/2}$ and the bound achieves by the blow up of $C_k$. For even cycles, we establish an upper bound with an extra constant factor. For odd cycles, we can only establish an upper bound with an extra factor depending on $k$. We prove that $ρ(P_{2l},m) \le \frac{m^l}{2(l-1)^{l-1}}$ and $ρ(P_{2l+1},m) \le \frac{m^{l+1}}{4l^l}$, where $l \ge 2$. We also conjecture the asymptotic value of $ρ(P_k, m)$. The entropy method is mainly used to prove our results.

Figures (4)

• Figure 1: Unbalanced blow up when $k=6$. The orange (resp. blue) parts have $\mu$ (resp. $\lambda$) vertices and $\lambda \mu \approx m/6$.
• Figure 2: The blow up of $P_k$ when $k$ is odd.
• Figure 3: The upper bound of the contribution of $e$ to $\sum_{j=1}^{2l}S_j^{+}$. The black nodes represent vertices in $J_e$, the white nodes represent vertices not in $J_e$ and gray nodes represent vertices which are not sure whether in $J_e$. The number on the side of each $v_j$ is an upper bound of the contribution of $e$ to $S_j^{+}$. The graph on the left shows when $e$ contributes $3/2$ to some $S_j^{+}$. At the upper left, we enumerate all possibilities of the gray nodes. The graph on the right shows the other case, split the cycle into "white node" paths of length $t$.
• Figure 4: A sample of capable $\{e_1,e_2,e_3\}$.

Theorems & Definitions (18)

• Conjecture 1.1: Pippenger and Golumbic pippenger1975inducibility
• Theorem 1.2: Alon alon1981number, Friegdut and Kahn friedgut1998number
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Conjecture 1.8
• Conjecture 1.9
• Conjecture 1.10