Maximizing Alternating Paths via Entropy

TL;DR

The paper settles the odd-length semi-inducibility problem for alternating paths in two-edge-colored graphs by proving an exact asymptotic upper bound on the homomorphism density t(P_{2k+1}^A,G) using the entropy method. Central to the approach is constructing an edge-colored forest H_{2k+1} with more edges than P_{2k+1}^A and relating their counts via entropy-based inequalities, including a detailed lower-bound construction and a nuanced upper-bound via local degree arguments. The main result is t(P_{2k+1}^A,G) ≤ k^k (k+1)^{k+1} (2k+1)^{-2k-1}, which is asymptotically tight, and the technique yields a robust framework for similar semi-inducibility problems. The authors also outline avenues for generalizing the method to broader classes of edge-colored trees, forests, and multi-color settings, suggesting rich future work.

Abstract

We prove that if $G$ is an $n$-vertex graph whose edges are coloured with red and blue, then the number of colour-alternating walks of length $2k+1$ with $k+1$ red edges and $k$ blue edges is at most $k^k(k+1)^{k+1}(2k+1)^{-2k-1}n^{2k+2}$. This solves a problem that was recently posed by Basit, Granet, Horsley, Kündgen and Staden. Our proof involves an application of the entropy method.

Figures (4)

• Figure 1: Two possible choices of the edge-coloured forest $H_3$ in Lemma \ref{['lem:key']}. Red edges are represented by solid lines and blue edges are represented by dashed lines.
• Figure 2: A representation of a possible choice of the edge-coloured forest $H_5$ in Lemma \ref{['lem:key']}. Each isolated edge is labelled by a number which represents the number of copies of that edge added in the construction. Each leaf that is not in an isolated edge, apart from $v_0$ and $v_5$, is labelled with a number. This number represents the number of copies of the unique path from that leaf to $\{v_0,\dots,v_5\}$ that are added in the construction of $H_5$ from $P_5^A$. For example, $H_5$ contains exactly 14 leaves which are adjacent to $v_3$ via a blue edge, and exactly 15 paths of length two starting with $v_2$ in which the first edge is red and the second edge is blue and the final vertex is not $v_4$.
• Figure 3: The edge-colored graph $H_{2k}$.
• Figure 4: A tree with edges of three colours.

Theorems & Definitions (31)

• Theorem 1.1: Basit et al. Basit+25+
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• proof : Proof of Theorem \ref{['thm:paths']}
• Definition 3.1
• Definition 3.2
• Lemma 3.3: Maximality of the Uniform Distribution
• Definition 3.4
• Definition 3.5