Resolving problems on the polynomial identity characterization of daisy cubes

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 Klavžar 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.

• The paper establishes that cube and distance polynomial identities uniquely characterize daisy cubes among partial cubes.
• It rigorously proves coefficientwise inequalities and shows that equality holds precisely when the graph is a daisy cube.
• It further identifies simplex graphs as the unique case where clique and distance polynomial upper bounds coincide, linking combinatorial and geometric features.

Characterization of Daisy Cubes via Cube Polynomial Identities

Introduction

The class of daisy cubes $Q_n(X)$, defined as induced subgraphs of the hypercube $Q_n$ formed by the intervals $I(x,0^n)$ for $x \in X \subseteq \{0,1\}^n$, generalizes notable graph families such as Fibonacci cubes and Lucas cubes. Their study is motivated not only by combinatorial and algebraic graph theory interests but also by connections to abstract simplicial complexes via the subset order relation. The structural understanding of daisy cubes, particularly their characterization among partial cubes via polynomial invariants, has been the subject of active research, with key open problems posed by Klavžar and Mollard [European J. Combin., 80 (2019) 214–223]. This essay synthesizes and assesses rigorous advances on the polynomial identity characterization of daisy cubes and related upper bounds for the cube polynomial in the context of partial cubes (2603.29577).

Polynomial Characterizations and Main Results

Let $G$ be a partial cube (i.e., an isometric subgraph of $Q_n$) and $u \in V(G)$. Three crucial graph polynomials are defined:

• Cube polynomial: $C_G(x) = \sum_{k \geq 0} c_k(G)x^k$, where $c_k(G)$ counts induced $k$-cubes in $Q_n$0.
• Distance polynomial: $Q_n$1, where $Q_n$2 counts vertices at distance $Q_n$3 from $Q_n$4.
• Distance cube polynomial: $Q_n$5, where $Q_n$6 counts induced $Q_n$7-cubes at distance $Q_n$8 from $Q_n$9.

The core contribution is the comprehensive resolution of the problem of whether certain polynomial equalities uniquely characterize daisy cubes. The main theorem rigorously establishes that for a partial cube $I(x,0^n)$0 and a distinguished vertex $I(x,0^n)$1 (typically taken as $I(x,0^n)$2), the following are equivalent:

1. $I(x,0^n)$3 is a daisy cube with $I(x,0^n)$4.
2. $I(x,0^n)$5.
3. $I(x,0^n)$6.
4. $I(x,0^n)$7.

Thus, the equality $I(x,0^n)$8 (and analogously, $I(x,0^n)$9) not only holds for daisy cubes (Proposition 2 in [SKM]), but in fact characterizes them. This directly answers two long-standing open problems posed in [SKM]. The paper further establishes that, in the more general context of arbitrary partial cubes, the inequalities $x \in X \subseteq \{0,1\}^n$0 and $x \in X \subseteq \{0,1\}^n$1 hold coefficientwise, with equality if and only if $x \in X \subseteq \{0,1\}^n$2 is a daisy cube and $x \in X \subseteq \{0,1\}^n$3.

Upper Bounds, Clique Polynomials, and Simplex Graphs

Beyond the daisy cube context, upper bounds on $x \in X \subseteq \{0,1\}^n$4 for partial cubes are explored. The paper revisits a result of Xie et al. [J. Graph Theory, 106 (2024) 907–922], showing $x \in X \subseteq \{0,1\}^n$5, where $x \in X \subseteq \{0,1\}^n$6 is the clique polynomial of the crossing graph $x \in X \subseteq \{0,1\}^n$7, whose vertices correspond to $x \in X \subseteq \{0,1\}^n$8-classes of $x \in X \subseteq \{0,1\}^n$9 and edges encode the crossing relation.

A significant structural outcome is the classification of graphs for which the two coefficientwise upper bounds coincide: $G$0 and $G$1. The paper proves that equality holds if and only if $G$2 is a simplex graph with $G$3 (i.e., the empty clique corresponds to $G$4), leveraging the known fact that simplex graphs are precisely the intersection of median graphs and daisy cubes. This draws a clear correspondence between combinatorial, algebraic, and geometric properties.

Technical Methods and Proof Highlights

The analysis uses structural characterizations of daisy cubes via maximal elements under the subset order and connects these to the intervals $G$5 forming full $G$6-cubes. A fundamental lemma exploits the correspondence between maximal vertices and intervals inducing cubes, offering an efficient test for the daisy-cube property. The proof strategy for the equivalence of polynomial identities is rooted in tracing the enumeration of subgraphs and distances under the isometric embedding of $G$7 into $G$8, systematically relating cube containment to algebraic expansions via the binomial theorem and partitioning contributions by top vertices.

The crossing graph and clique polynomial bounds are linked via a convex-hull construction (as in [xfx24]), characterizing the extremal case through closure operators and embedding arguments relating $G$9 to its median hull.

Implications and Future Directions

These characterizations resolve key open questions about the polynomial invariants of daisy cubes and unify several strands in the structural theory of partial cubes, specifically in the algebraic combinatorics of subgraph enumeration. The new equivalence theorems place the cube polynomial at the center of daisy cube recognition, enabling purely algebraic criteria for their identification. This facilitates the development of recognition algorithms and the transfer of techniques across combinatorial, geometric, and algebraic frameworks.

The identification of simplex graphs via the coincidence of clique and distance polynomials hints at a deeper interplay between cube structure, clique theory, and metric properties in partial cubes. Potential future work includes:

• Extending similar characterizations to broader classes of isometric subgraphs in higher-dimensional lattices,
• Investigating algorithmic implications for efficient recognition and parameter computation,
• Exploring connections with topological combinatorics via the abstraction to simplicial complexes,
• Applying these polynomial invariants in chemical graph theory and network science contexts where partial cubes naturally arise.

Conclusion

The rigorous algebraic characterization of daisy cubes within partial cubes via cube and distance polynomial identities establishes necessary and sufficient conditions that resolve foundational questions in the field (2603.29577). The results synthesize geometric, combinatorial, and algebraic perspectives and set the stage for broader investigation into polynomial invariants and their structural implications for subgraph-closed classes in discrete mathematics.