The degree-diameter problem for plane graphs with pentagonal faces

TL;DR

This work solves the degree-diameter problem for pentagulations, i.e., plane graphs whose every face is a 5-cycle, at diameter $d=3$. By a detailed cycle-structure analysis—eliminating triangles, characterizing separating cycles, and introducing dislocated 4-cycles alongside HI/I subgraphs—the authors prove that for maximum degree $Δ ≥ 8$, the order satisfies $n(Δ,3) ≤ 3Δ - 1$; sharpness holds for odd $Δ$. The result completes the exact degree-diameter characterization for diameter-3 plane graphs with uniform face length, complementing known bounds for triangulations ($ρ=3$) and quadrangulations ($ρ=4$). The methods combine geometric and combinatorial arguments on cycle domination, region subdivision, and disjoint-4-cycle configurations, providing a template for analogous problems on other face-pattern families.

Abstract

The degree-diameter problem consists of finding the maximum number of vertices $n$ of a graph with diameter $d$ and maximum degree $Δ$. This problem is well studied, and has been solved for plane graphs of low diameter in which every face is bounded by a 3-cycle (triangulations), and plane graphs in which every face is bounded by a 4-cycle (quadrangulations). In this paper, we solve the degree diameter problem for plane graphs of diameter 3 in which every face is bounded by a 5-cycle (pentagulations). We prove that if $Δ\geq 8$, then $n \leq 3Δ- 1$ for such graphs. This bound is sharp for $Δ$ odd.

Figures (33)

• Figure 1: Some steps in the proof of Lemma \ref{['lem_5_3_1']}.
• Figure 2: A 4-cycle dominating its interior which has $k=2$ paths of length 3 and $k-1=1$ paths of length 2 between two non-cycle-adjacent vertices $v_1$ and $v_3$, illustrating Theorem \ref{['thm:4_cycle_description']}.
• Figure 3: Two diameter three pentagulations that contain 4-cycles, $\mathcal{H}$ and $\mathcal{I}$. Pairs of non-adjacent grey vertices dominate regions bounded by bold 4-cycles.
• Figure 4: In $G$, there is no pair of dislocated 4-cycles. In $H$, any pair of 4-cycles in which both cycles dominate their interior or exterior is dislocated.
• Figure 5: The graphs $\mathcal{H}$ and $\mathcal{I}$, with the labels used in the proofs of Lemmas \ref{['lem_H_as_subgraph']} and \ref{['lem_I_as_subgraph']}.

Theorems & Definitions (56)

• Remark 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4