Topological Properties of Benzenoid Systems. XXI. Theorems, Conjectures, Unsolved Problems

The main known mathematical results (in the form of 32 theorems and 5 conjectures) about benzenoid systems are collected. A few new results (seven theorems) are proved. Seven unsolved problems are also pointed out. The paper contains results on the basic properties of benzenoid graphs, on the number of Kekule structures and on Clar\u27s resonant sextet formulas

Planar k-cycle resonant graphs with k=1,2

AbstractA connected graph is said to be k-cycle resonant if, for 1⩽t⩽k, any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M-alternating cycle. The concept of k-cycle resonant graphs was introduced by the present authors in 1994. Some necessary and sufficient conditions for a graph to be k-cycle resonant were also given. In this paper, we improve the proof of the necessary and sufficient conditions for a graph to be k-cycle resonant, and further investigate planar k-cycle resonant graphs with k=1,2. Some new necessary and sufficient conditions for a planar graph to be 1-cycle resonant and 2-cycle resonant are established

A Maximum Resonant Set of Polyomino Graphs

A polyomino graph $H$ is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. In this paper, we show that if $K$ is a maximum resonant set of $H$, then $H-K$ has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to its Clar number. Based on this result, we have that the maximum forcing number of a polyomino graph can be computed in polynomial time. We also show that if $K$ is a maximal alternating set of $H$, then $H-K$ has a unique perfect matching.Comment: 13 pages, 6 figure

Resonance graphs of plane bipartite graphs as daisy cubes

We characterize all plane bipartite graphs whose resonance graphs are daisy cubes and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if $G$ is a plane elementary bipartite graph other than $K_2$, then the resonance graph $R(G)$ is a daisy cube if and only if the Fries number of $G$ equals the number of finite faces of $G$, which in turn is equivalent to $G$ being homeomorphically peripheral color alternating. Next, we extend the above characterization from plane elementary bipartite graphs to all plane bipartite graphs and show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if $G$ is weakly elementary bipartite and every elementary component of $G$ other than $K_2$ is homeomorphically peripheral color alternating. Along the way, we prove that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes

An Easy Combinatorial Algorithm for the Construction of Sextet Polynomials of Cata-Condensed Benzenoid Hydrocarbons

Two types oif i>mning techniques for cer.tain trees are described and utilized into an easy combinatorial a1gorithm for the f:?YStematic construction o.f sextet polynomials of catacon.densed benzenoid hydrocarbons of large sizes. The a1gori.thm oiffers an ailte!l\u27na.tive to ex~sting methods !or the enumeration of Kekule structures which is not restricted to non-branched systems

An Easy Combinatorial Algorithm for the Construction of Sextet Polynomials of Cata-Condensed Benzenoid Hydrocarbons

Two types oif i>mning techniques for cer.tain trees are described and utilized into an easy combinatorial a1gorithm for the f:?YStematic construction o.f sextet polynomials of catacon.densed benzenoid hydrocarbons of large sizes. The a1gori.thm oiffers an ailte!l\u27na.tive to ex~sting methods !or the enumeration of Kekule structures which is not restricted to non-branched systems