Resolving problems on the polynomial identity characterization of daisy cubes

TL;DR

This paper characterizes daisy cubes among partial cubes using polynomial equalities, resolving open problems and establishing bounds related to cube and clique polynomials.

Contribution

It provides necessary and sufficient conditions for a partial cube to be a daisy cube via polynomial equalities, addressing open questions in the field.

Findings

Characterization of daisy cubes via polynomial equalities.

Proof that certain polynomial conditions are equivalent to being a daisy cube.

Establishment of bounds for cube and clique polynomials and their comparison.

Abstract

Let $X\subseteq\{0,1\}^n$ be a set of binary strings of length $n$. The daisy cube $Q_n(X)$ is the subgraph of the hypercube $Q_n$ induced by the union of the intervals $I(x,0^n)$ for $x\in X$. As a subclass of partial cubes, it generalizes Fibonacci cubes and Lucas cubes. For a graph $G$ and a vertex $u\in V(G)$, we consider the cube polynomial $C_G(x)$, the distance cube polynomial $D_{G,u}(x,y)$, and the polynomial $W_{G,u}(x)$, which count $k$-cubes, $k$-cubes at distance from $u$, and vertices at distance $k$ from $u$, respectively. In this paper, we prove that for a partial cube $G$ with a vertex $u\in V(G)$, $G$ is a daisy cube and $u=0^n$ if and only if one of the following equivalent conditions holds: (1) $C_{G}(x)=W_{G,u}(x+1)$; (2) $D_{G,u}(x,y)=W_{G,u}(x+y)$; (3) $D_{G,u}(x,y)=C_{G}(x+y-1)$. In particular, conditions (1) and (3) give affirmative answers to two open problems posed by Klav\v{z}ar and Mollard [European J. Combin., 80 (2019) 214--223]. Further, we obtain that for arbitrary partial cube $G$, $D_{G,u}(x,y)\leq W_{G,u}(x+y)$ and $C_{G}(x)\leq W_{G,u}(x+1)$. Besides, another bound for $C_G(x)$ due to Xie et al. [J. Graph Theory, 106 (2024) 907--922] is given by the clique polynomial $Cl_{G^\#}(x+1)$ of the crossing graph of $G$. We also compare these two bounds and show that the simplex graphs form the unique class of graphs for which the two bounds coincide.