Cubicity, Degeneracy, and Crossing Number

A $k$-box $B=(R_1,…,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times …\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as $\cubi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/ 0607092, 2006.] that, for a graph $G$ with maximum degree $\Delta$, $\cubi(G)\leq \lceil 4(\Delta +1)\log n\rceil$. In this paper, we show that, for a $k$-degenerate graph $G$, $\cubi(G) \leq (k+2) \lceil 2e \log n \rceil$. Since $k$ is at most $\Delta$ and can be much lower, this clearly is a stronger result. This bound is tight. We also give an efficient deterministic algorithm that runs in $O(n^2k)$ time to output a $8k(\lceil 2.42 \log n\rceil + 1)$ dimensional cube representation for $G$. An important consequence of the above result is that if the crossing number of a graph $G$ is $t$, then $\boxi(G)$ is $O(t^{1/4}{\lceil\log t\rceil}^{3/4})$ . This bound is tight up to a factor of $O((\log t)^{1/4})$. We also show that, if $G$ has $n$ vertices, then $\cubi(G)$ is $O(\log n + t^{1/4}\log t)$. Using our bound for the cubicity of $k$-degenerate graphs we show that cubicity of almost all graphs in $\mathcal{G}(n,m)$ model is $O(d_{av}\log n)$, where $d_{av}$ denotes the average degree of the graph under consideration.

💡 Research Summary

The paper investigates two geometric graph parameters—boxicity (the minimum dimension needed to represent a graph as an intersection of axis‑parallel boxes) and cubicity (the same but with unit‑length boxes, i.e., cubes). Building on earlier work that bounded cubicity by O(Δ log n) where Δ is the maximum degree, the authors exploit the finer structural measure of k‑degeneracy (every vertex has at most k forward neighbors in some vertex ordering).

The main theorem proves that any k‑degenerate graph G on n vertices satisfies
 cubicity(G) ≤ (k + 2)·⌈2e log n⌉,
and that this bound is asymptotically tight. The proof proceeds in two parts. First, a probabilistic coloring scheme uses k + 2 colors and repeats ⌈2e log n⌉ independent random colorings. For each non‑edge (x, y) (with x preceding y in the degeneracy order) a “strong support set” T_xy is defined; the probability that T_xy receives a proper coloring in at least one of the repetitions is shown to exceed 1, guaranteeing that every non‑edge is “good” in some coloring. Second, each good coloring is turned into a unit‑interval graph I_{i,j} by assigning to every vertex a unit interval whose left endpoint depends on its color and on the maximum index of a neighbor sharing that color. The intersection of all such I_{i,j} exactly equals G, and by Roberts’ lemma this yields the cubicity bound.

Beyond the existential proof, the authors present a deterministic algorithm that constructs the required colorings in O(n k) time and then builds the interval representations, resulting in an explicit (8k·(⌈2.42 log n⌉ + 1))-dimensional cube representation in O(n²k) time. This is the first polynomial‑time algorithm that achieves a dimension matching the theoretical bound up to constant factors.

The paper then connects these results to the crossing number CR(G) = t, the minimum number of edge crossings over all planar drawings of G. By partitioning G into t planar subgraphs and applying the degeneracy‑based bound to each, the authors derive
 boxicity(G) = O(t^{1/4} (log t)^{3/4})
and consequently
 cubicity(G) = O(log n + t^{1/4} log t).
Both bounds are shown to be tight up to polylogarithmic factors.

Finally, the authors consider the random graph model G(n,m). Almost every graph in this model has average degree d_av = 2m/n and is O(d_av)-degenerate. Plugging this into the main theorem yields
 cubicity(G) = O(d_av log n)
for almost all graphs, improving on earlier results that only bounded boxicity in the same order.

In summary, the paper makes four significant contributions: (1) a tight O((k+2) log n) bound on cubicity for k‑degenerate graphs, (2) a deterministic O(n²k) algorithm producing an explicit low‑dimensional cube representation, (3) new bounds linking crossing number to boxicity and cubicity, and (4) an application showing that almost all random graphs have cubicity O(d_av log n). These results deepen the understanding of how subtle structural parameters such as degeneracy influence the geometric complexity of graph representations.