2-extendability of (4,5,6)-fullerenes

TL;DR

The paper resolves which $(4,5,6)$-fullerenes are $2$-extendable by combining a forbidden-subgraph approach with Tutte's theorem on perfect matchings. The authors identify exactly the non-$2$-extendable graphs as the four sporadic fullerenes $F_{12},F_{14},F_{18},F_{20}$ and five infinite classes, and they show that all fullerenes with anti-Kekulé number $3$ are non-$2$-extendable, with additional results that such non-$2$-extendable examples exist for arbitrarily large even numbers of vertices. The analysis relies on a patch-growth framework and a comprehensive catalog of forbidden subgraphs $H_i$, linking local obstructions to global extendability. These findings advance understanding of the interplay between fullerene structure and perfect-matchings, with implications for chemical stability notions related to Kekulé structures.

Abstract

A (4,5,6)-fullerene is a plane cubic graph whose faces are only quadrilaterals, pentagons and hexagons, which includes all (4,6)- and (5,6)-fullerenes. A connected graph $G$ with at least $2k+2$ vertices is $k$-extendable if $G$ has perfect matchings and any matching of size $k$ is contained in a perfect matching of $G$. We know that each (4,5,6)-fullerene graph is 1-extendable and at most 2-extendable. It is natural to wonder which (4,5,6)-fullerene graphs are 2-extendable. In this paper, we completely solve this problem (see Theorem 3.3): All non-2-extendable (4,5,6)-fullerenes consist of four sporadic (4,5,6)-fullerenes ($F_{12},F_{14},F_{18}$ and $F_{20}$) and five classes of (4,5,6)-fullerenes. As a surprising consequence, we find that all (4,5,6)-fullerenes with the anti-Kekulé number 3 are non-2-extendable. Further, there also always exists a non-2-extendable (4,5,6)-fullerene with arbitrarily even $n\geqslant10$ vertices.

Figures (15)

• Figure 1: Illustration for a (4,5,6)-fullerene $T_3$ in $\mathcal{T}$, layer and traversed edge.
• Figure 2: (4,5,6)-fullerenes $F_{12}, F_{14}, F_{18}$ and $F_{20}$ are non-2-extendable.
• Figure 3: The possible subgraphs $H_i$($1\leqslant i\leqslant 64$).
• Figure 4: The possible subgraphs $H_i$, $65\leqslant i\leqslant 122$.
• Figure 5: Illustration for Subcase 1.1.

Theorems & Definitions (15)

• Lemma 2.1: 8
• Lemma 2.2: 8
• Lemma 2.3: 8
• Lemma 2.4: 3
• Lemma 2.5
• Theorem 2.6: 5
• Theorem 3.1: 9
• Lemma 3.2
• proof
• Theorem 3.3