On the anti-Kekulé problem of cubic graphs

Abstract

An edge set S of a connected graph G is called an anti-Kekulé set if G−S is connected and has no perfect matchings, where G − S denotes the subgraph obtained by deleting all edges in S from G. The anti-Kekulé number of a graph G, denoted by ak(G), is the cardinality of a smallest anti-Kekulé set of G. It is NP-complete to find the smallest anti-Kekulé set of a graph. In this paper, we show that the anti-Kekulé number of a 2-connected cubic graph is either 3 or 4, and the anti-Kekulé number of a connected cubic bipartite graph is always equal to 4. Furthermore, a polynomial time algorithm is given to find all smallest anti-Kekulé sets of a connected cubic graph.

The anti-Kekulé number of graphs

“…Claim 1: G � − S has no even components and it has exactly |S| + 2 odd components. Next we prove the main result of this subsection due to Li et al [22].…”

“…If the graphs being considered are bipartite, then a stronger result can be obtained by using Hall's Theorem. Theorem 41 [22] If G is either a toroidal fullerene or a bipartite Klein-bottle fullerene, then ak(G) = 4.…”

“…Recall that a connected regular graph of degree three is called cubic graph. Recently, Li et al [22] studied the anti-Kekulé problem of cubic graphs.…”

“…Li et al [22] showed that a 2-connected cubic graph has anti-Kekulé number either 3 or 4. This arises the following problem.…”

Anti-Kekulé number of the {(3, 4), 4}-fullerene*

“…In general, Li et al (2019) showed that the anti-Kekulé number of a 2-connected cubic graph is either 3 or 4; moreover, all (4,6)fullerenes have the anti-Kekulé number 4, and all the (3,6)-fullerenes have anti-Kekulé number 3. Zhao and Zhang (2020) confirmed all (4,5,6)-fullerenes have anti-Kekulé number 3, which consist of four sporadic (4,5,6)-fullerenes (F 12 , F 14 , F 18 , and F 20 ) and three classes of (4,5,6)-fullerenes with at least two and at most six pentagons.…”