[Paper Review] Number of Matchings of Low Order in (4,6)-Fullerene Graphs

Zhi-Feng Wei, Heping Zhang|arXiv (Cornell University)|Jan 22, 2023

Fullerene Chemistry and Applications9 citations

TL;DR

The paper provides corrected formulae for numbers of 4-, 5-, and 6-matchings in (4,6)-fullerene graphs, distinguishing graph types (cube, tubular, lantern, dispersive) and expressing counts in terms of hexagonal faces and dual-squares.

ABSTRACT

We obtain the formulae for the numbers of 4-matchings and 5-matchings in terms of the number of hexagonal faces in (4, 6)-fullerene graphs by studying structural classification of 6-cycles and some local structural properties, which correct the corresponding wrong results published. Furthermore, we obtain a formula for the number of 6-matchings in tubular (4, 6)-fullerenes in terms of the number of hexagonal faces, and a formula for the number of 6-matchings in the other (4,6)-fullerenes in terms of the numbers of hexagonal faces and dual-squares.

Motivation & Objective

• Classify 6-cycles in (4,6)-fullerene graphs and establish structural types (cube, tubular, lantern, dispersive).
• Derive accurate recurrence relations linking low-order matchings to higher-order subgraphs.
• Provide explicit formulas for the numbers of 4-, 5-, and 6-matchings in terms of hexagonal faces and dual-squares.

Proposed method

• Classify 6-cycles: hexagonal facial, dual-square, square-cap, or square-cap with hexagon-layers (Theorem 2.7).
• Derive local structural properties and global graph-type implications (lantern, dispersive, tubular).
• Set up recurrence relations connecting M(G,k) for k=4,5,6 to counts of specific subgraphs (N(·)) and structural parameters (h, y).
• Compute subgraph counts N(·) (e.g., N(K), N(Q), N(S), N(R)) and express M(G,4), M(G,5), M(G,6) by linear recurrences.
• Provide closed-form formulas for M(G,4), M(G,5); and piecewise formulas for M(G,6) depending on graph class (Theorem 3.12).
• Enumerate auxiliary subgraph counts (Lemmas 3.1–3.11) to support recurrences (Corollaries 3.6–3.13).

Experimental results

Research questions

• RQ1 What are the correct structural possibilities for 6-cycles in (4,6)-fullerenes and how do they affect matching counts?
• RQ2 How can one derive accurate low-order matching counts (4-, 5-, 6-matchings) for all (4,6)-fullerenes using graph structure parameters?
• RQ3 Do unified closed-form expressions exist for 6-matchings across all (4,6)-fullerenes or must they be separated by graph families?
• RQ4 How do counts of hexagonal faces and dual-squares control the numbers of low-order matchings?
• RQ5 What corrections are needed to previous erroneous formulas for 4- and 5-matchings in this graph class?

Key findings

• The 6-cycle structure in (4,6)-fullerenes falls into four types: hexagonal facial cycle, dual-square, square-cap, or square-cap with hexagon-layers (Theorem 2.7).
• There is no single unified formula for 6-matchings across all (4,6)-fullerenes; two formulae are given depending on tubular vs. non-tubular (Theorem 3.12).
• Exact formulas: M(G,4) = (27/8)h^4 + (81/4)h^3 + (273/8)h^2 + (117/4)h + 9 (3.4a) and M(G,5) = (81/40)h^5 + (27/4)h^4 - (9/8)h^3 + (39/4)h^2 - (27/5)h (3.4b).
• For tubular (G ∈ T), M(G,6) = (81/80)h^6 - (81/80)h^5 - (99/16)h^4 + (405/16)h^3 - (1873/40)h^2 + (1231/30)h - 9 (Theorem 3.12).
• For non-tubular (G ≠ T, cube excluded) with y dual-squares: M(G,6) = (81/80)h^6 - (81/80)h^5 - (99/16)h^4 + (405/16)h^3 - (1873/40)h^2 + (407/10)h - 16 + y (Theorem 3.12).
• The paper corrects prior incorrect counts for 4- and 5-matchings and provides corrected N(P) and related subgraph counts (Remark 3.3; Lemmas 3.5–3.11).