Strong log-convexity of genus sequences

TL;DR

This work addresses whether the genus distribution $\{a_g(G)\}$ of every graph $G$ is log-concave, a long-standing conjecture. By constructing explicit $4$-connected graphs $G_{g,k}$ of genus $g$ using stacked antiprisms connected by antiprismatic paths, the author derives tight lower and upper bounds on embedding counts $a_{g+r}(G_{g,k})$ and demonstrates that the alternating sequence $a_{g+1},a_{g+3},\dots,a_{g+2k-1}$ is strictly log-convex for large parameters, thereby refuting the conjecture in a strong form. The results extend to show the existence of $k$ consecutive strictly log-convex terms and reveal that, despite the failure of log-concavity, genus distributions remain unimodal; this sharpens the understanding of map enumeration on surfaces and its connections to $\Delta$-matroids and related polynomials. The paper thus provides a definitive negative answer to LCGD and highlights nuanced growth behaviors in topological graph embeddings.

Abstract

For a graph $G$, and a nonnegative integer $g$, let $a_g(G)$ be the number of $2$-cell embeddings of $G$ in an orientable surface of genus $g$ (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker [Genus distributions for bouquets of circles, J. Combin. Theory Ser. B 47 (1989), 292-306] proposed a conjecture that the sequence $a_0(G),a_1(G),a_2(G),\dots$ is log-concave for every graph $G$. This conjecture is reminiscent to the Heron-Rota-Welsh Log Concavity Conjecture that was recently resolved in the affirmative by June Huh et al., except that it is closer to the notion of $Δ$-matroids than to the usual matroids. In this short paper, we disprove the Log Concavity Conjecture of Gross, Robbins, and Tucker by providing examples that show strong deviation from log-concavity at multiple terms of their genus sequences.

Figures (5)

• Figure 1: (a) The antiprismatic path of length 9 gives the antiprism $\widehat{C}_9$ after identifying the left edge $v_0^0 v_0^1$ with the one on the right. (b) The stacked antiprism $\widehat{C}_9^4$ of width 9 and height 4. (c) The genus 0 cylinder $H_m^{k,b,d}$ with parameters $m=9$, $k=2$, $b=2$, and $d=3$. The vertices and edges on the left are identified with the vertices and edges on the right side to give a planar graph on the cylinder.
• Figure 2: (i) Replacing a triangular face by the triangular grid of side length $l$, illustrated here with $l=7$. (ii) Replacing four triangles inside the triangular grid so that a face with all vertices of degree 4 is obtained.
• Figure 3: Merging three big faces into a single face $F$.
• Figure 4: Triangles on a connector (drawn horizontally). The three black vertices belong to $V_D$.
• Figure 5: Degrees of freedom when the homotopy of unstable triangles is changed. The shaded triangles are facial (with the shown or with the opposite orientation on the surface).

Theorems & Definitions (20)

• Conjecture 1: Gross, Robbins, and Tucker, 1989
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• Lemma 8