Structure and enumeration results of matchable Lucas cubes

Abstract

A lucasene is a hexagon chain that is similar to a fibonaccene, an L-fence is a poset the Hasse diagram of which is isomorphic to the directed inner dual graph of the corresponding lucasene. A new class of cubes, which named after matchable Lucas cubes according to the number of its vertices (or elements), are a series of directed or undirected Hasse diagrams of filter lattices of L-fences. The basic properties and several classes of polynomials, e.g. rank generating functions, cube polynomials and degree sequence polynomials, of matchable Lucas cubes are obtained. Some special conclusions on binomial coefficients and Lucas triangle are given.A set P equipped with a binary relation ≤ satisfying reflexivity, antisymmetry and transitivity is said to be a partially ordered set (poset for short). Given any poset P , the dual P * of P is formed by defining x ≤ y to hold in P * if and only if y ≤ x holds in P . A subposet S of P is a chain if any two elements of S are comparable, and denoted by n if |S| = n [3]. Let x ≺ y denote y cover s x in P , i.e. x < y and. The set of all filters of a poset P is denoted by F (P ), and carries the usual anti-inclusion order, forms a finite distributive lattice [3,31] called filter lattice. For a finite lattice L, we denote by0 L (resp.1 L ) the minimum (resp. maximum) element in L.The symmetric difference of two finite sets A and B is defined asis a perfect matching of a graph and C is an M -alternating cycle of the graph, then the symmetric difference of M and edge-set E(C) is another perfect matching of the graph, which is simply denoted by M ⊕ C. Let G be a plane bipartite graph with a perfect matching, and the vertices of G are colored properly black and white such that the two ends of every edge receive different colors. An M -alternating cycle of G is said to be proper, if every edge of the cycle belonging to M goes from white end-vertex to black end-vertex by the clockwise orientation of the cycle; otherwise improper [43]. An inner face of a graph is called a cell if itsboundary is a cycle, and we will say that the cycle is a cell too. For some concepts and notations not explained in the paper, refer to [3,8,31] for poset and lattice, [1,11] for graph theory. Zhang and Zhang [45] extended the concept of Z-transformation graph of a hexagonal system to plane bipartite graphs. Definition 2.1 ([45]) Let G be a plane bipartite graph. The Z-transformation graph Z(G) is defined on M(G): M 1 , M 2 ∈ M(G) are joined by an edge if and only if M 1 ⊕ M 2 is a cell of G. And Z-transformation digraph Z(G) is the orientation of Z(G): an edge M 1 M 2 of Z(G) is oriented from M 1 to M 2 if M 1 ⊕ M 2 form a proper M 1 -alternating (thus improper M 2 -alternating) cell. Let G be a bipartite graph, from Theorem 4.1.1 in [21], we have that G is elementary if and only if G is connected and every edge of G lies in a perfect matching of G. Let G be a plane bipartite graph with a perfect matching, a binary relation ≤ on M(G) is defined as: for M 1 , M 2 ∈ M(G), M 1 ≤ M 2 if and only if Z(G) has a d...

The (non-)existence of perfect codes in Lucas cubes

“…Lucas cubes, introduced in [17], have attracted the attention as well due to the fact that these cubes are closely related to the Fibonacci cubes. They have also been widely studied [3,4,6,11,13,14,18,20,23].…”

Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

“…The Fibonacci cube is not the only interesting graph that arises as the braid graph for a reduced expression. For instance, the matchable Lucas cubes introduced [18] can be found in the Coxeter system of type D 5 as seen in the following example. The matchable Lucas cubes are very similar to the Fibonacci cubes, for example, the number of vertices in the n-th matchable Lucas cube is equal to the n-th number L n in the sequence of Lucas numbers defined by L n = L n−1 + L n−2 with initial conditions L 0 = 2 and L 1 = 1.…”

“…Consider the reduced expression 4534313234313 for w ∈ W (D 5 ) from Example 6.4. The braid graph, depicted in Figure 10(c), is isomorphic to the 6th matchable Lucas cube as introduced in [18]. In fact, it is not hard to see that the braid graph is a gluing together of the braid graphs for the reduced expressions 45343132343 and 453431323.…”

Binary Coding of Resonance Graphs of Catacondensed Polyhexes

“…A similar approach was applied in [24] where the so-called matchable Lucas cube was introduced. It is also worth mentioning that resonance graphs of kinky benzenoid systems belong to a class of isometric subgraphs of hypercubes called daisy cubes [33].…”