Bipartite-Hole-Number: Graph Invariant

• Bipartite-hole-number is defined as the maximum integer r such that for every split r = s + t, the graph contains disjoint vertex sets of sizes s and t with no connecting edges.
• It underpins extremal graph theory by influencing Hamiltonicity, pancyclicity, and cycle conditions, thereby extending traditional measures like the independence number.
• Computing the bipartite-hole-number is NP-complete in general, though efficient algorithms exist for fixed parameters, making it a key focus in modern graph theory research.

The bipartite-hole-number of a graph, denoted $\widetilde{\alpha}(G)$ (also written $\widetilde\alpha(G)$), is a structural invariant that generalizes the independence number and governs the presence of large bipartite "holes"—that is, pairs of disjoint vertex sets with no edges between them. This parameter plays a decisive role in extremal graph theory, particularly in Hamiltonicity, pancyclicity, and related degree conditions, and is a focal point in several recent extensions of classical theorems.

1. Formal Definition and Characterizations

Let $G = (V, E)$ be a simple graph of order $n = |V|$. An $(s, t)$–bipartite-hole in $G$ is a pair of disjoint subsets $S, T \subseteq V$ with $|S| = s$ and $|T| = t$, such that there are no edges between $S$ and $T$: $E(S, T) = \emptyset .$ The bipartite-hole-number $\widetilde\alpha(G)$ is the maximum integer $r$ such that for every pair of non-negative integers $s, t$ with $s + t = r$, the graph $G$ contains such an $(s, t)$–bipartite hole: $\widetilde\alpha(G) = \max\left\{\, r : \text{for all } s, t \ge 0,\, s + t = r,\, \text{%%%%17%%%%–bipartite-hole exists in } G \,\right\} .$ Equivalently, it is the minimal $k$ such that no pair of disjoint sets of size $s, t$ with $s + t = k+1$ are nonadjacent. Thus, for all $r < \widetilde\alpha(G)+1$, every partition of $r$ into $(s, t)$ yields a bipartite hole, but for $k = \widetilde\alpha(G)+1$ there is at least one $(s, t)$ missing.

For bipartite graphs $(A \cup B, E)$, related variants include the maximal $k$ for which there always exists a pair $X \subseteq A$, $Y \subseteq B$ with $|X| = |Y| = k$ and no edges between $X$ and $Y$—in the complement, this is a $K_{k, k}$ subgraph.

2. Computation, Algorithms, and Complexity

Determining $\widetilde\alpha(G)$ for an arbitrary graph $G$ is computationally intractable in general due to its close relationship with the Maximum Balanced Biclique problem. The decision problem

$$\text{BHN: Given } (G, k), \text{ is } \widetilde\alpha(G) \geq k?$$

is NP-complete, as shown by reduction from the Balanced Complete Bipartite Subgraph problem (McDiarmid et al., 2016). Furthermore, under ETH-type complexity assumptions, $\widetilde\alpha(G)$ cannot be approximated within any factor $2^{(\log n)^{\delta}}$ for some $\delta > 0$.

Nevertheless, for fixed $k$ and fixed $(s, t)$, checking whether an $(s, t)$–bipartite hole exists reduces to searching for a $K_{s,t}$ in the complement $G^c$, which is polynomial-time for fixed parameters.

A constructive result is an $O(n^3)$-time algorithm that either outputs a Hamilton cycle in $G$ or provides a certificate that $\widetilde\alpha(G) > \delta(G)$ (McDiarmid et al., 2016).

3. Extremal Examples and Range

The extremes of $\widetilde\alpha(G)$ are:

Graph	$\widetilde\alpha(G)$	Remarks
Complete $K_n$	$1$	All vertex pairs connected
Edgeless $\overline{K_n}$	$n-1$	All nontrivial bipartitions form bipartite holes
Complete bipartite $K_{a,b}$ ($a \le b$)	$2a$	Worst-case: both sets lie in smaller part
Path $P_n$, $n \geq 3$	$2$	No $(2,2)$–hole once $n$ is large

In general, for any $G$, $\alpha(G) \leq \widetilde\alpha(G) \leq n-1$, and $\widetilde\alpha(G) \leq \chi(G)$, where $\chi(G)$ is the chromatic number (Li et al., 11 Jun 2025, Ellingham et al., 1 Nov 2025).

4. Relationships to Other Graph Invariants

The bipartite-hole-number generalizes the independence number $\alpha(G)$. Every large independent set $I$ of size $\alpha(G)$ and any outside vertex $v$ gives a $(\alpha(G), 1)$–bipartite hole, hence $\alpha(G) \leq \widetilde\alpha(G)$.

It is also linked to vertex-connectivity: $\kappa(G) + \widetilde\alpha(G) \leq n$ where $\kappa(G)$ is the vertex-connectivity. This follows from the extremal expressions (Ellingham et al., 1 Nov 2025, Cheng et al., 20 Nov 2025): $$\widetilde\alpha(G) = n - \max_{1 \leq s \leq n - \delta(G)} \min_{|S|=s} |N(S)|$$ where $N(S)$ is the set of neighbors of $S$ in $G$. This duality highlights the parameter's role as a measure of global nonexpansion.

5. Fundamental Extremal Results and Hamiltonicity

Minimum Degree and Hamiltonicity

The central theorem of McDiarmid–Yolov states (McDiarmid et al., 2016, Ellingham et al., 1 Nov 2025, Cheng et al., 20 Nov 2025):

Let $G$ be a graph of order $n \geq 3$. If $\delta(G) \geq \widetilde\alpha(G)$, then $G$ is Hamiltonian.

This extends Dirac's classical theorem ($\delta(G) \ge n/2$ ensures Hamiltonicity), since any graph with $\delta(G) \ge n/2$ must have $\widetilde\alpha(G) \le n/2$.

Degree Sum and Cyclability

Ore-type and cyclability extensions include:

For 2-connected $G$, if $d(x) + d(y) \geq 2\widetilde\alpha(G) - 1$ for all nonadjacent $x, y$, then $G$ is Hamiltonian, except for a family of exceptional graphs (Ellingham et al., 1 Nov 2025, Cheng et al., 20 Nov 2025).

Moreover, every (not necessarily Hamiltonian) graph contains a cycle through all vertices $v$ with $d_G(v) \geq \widetilde\alpha(G)$ (Li et al., 11 Jun 2025, Ellingham et al., 1 Nov 2025).

Pancyclicity

Correia, Draganic, and Sudakov (2024) established:

If $\delta(G) \geq \widetilde\alpha(G)$, then $G$ is pancyclic unless $G \cong K_{n/2, n/2}$ (Ellingham et al., 1 Nov 2025).

This result unifies and sharpens various cycle-extremal thresholds, showing that large bipartite holes obstruct not only Hamiltonicity but the presence of cycles of all lengths.

6. Bipartite-Hole-Number in Bipartite Graphs

For balanced bipartite graphs $(A \cup B, E)$, the function $f(n, \Delta)$ denotes the largest integer $k$ such that every $n \times n$ bipartite $G$ with $\Delta(G|_A) \leq \Delta$ contains a $k \times k$ bi-hole (i.e., $K_{k,k}$ in the complement) (Axenovich et al., 2020):

• For large $\Delta$, $$f(n, \Delta) = \Theta\left(\frac{\log\Delta}{\Delta} n\right)$$.
• Exact values and tight bounds are known for small $\Delta$, e.g., $f(n, 2) = \lfloor n/2 \rfloor - 1$.
• The case $\Delta=3$ remains open within a substantial constant gap.

Degree-based lower bounds, such as Caro–Wei analogues, state that for $G$ with degree sequence $d(v)$,

$$\text{bi-hole-number} \geq \left\lfloor \frac{1}{2} \sum_{v \in V(G)} \frac{1}{d(v)+1} \right\rfloor$$

(Kogan, 2020, Ehard et al., 2020). There are refined bounds for higher degeneracy and average degree, supporting optimal bihole size under various constraints.

7. Open Problems and Research Directions

Several open questions are actively investigated:

• Hamilton-connectedness and cycle lengths: Whether the degree-sum conditions with $\widetilde\alpha(G)$ can be further sharpened to ensure Hamilton-connectedness or pancyclicity (beyond known exceptions) (Ellingham et al., 1 Nov 2025, Cheng et al., 20 Nov 2025).
• Asymptotics for $f(n, \Delta)$ in bipartite graphs: Closing the constant gap for the case $\Delta=3$ and determining sharp thresholds for larger $\Delta$ (Axenovich et al., 2020, Ehard et al., 2020).
• Computation and approximation: Determining whether more efficient algorithms or better approximation factors can be obtained for general graphs (McDiarmid et al., 2016).
• Connectivity versus $\widetilde\alpha(G)$: Further exploring direct relationships between $\widetilde\alpha(G)$ and other critical graph invariants (e.g., vertex-connectivity, chromatic number) (Ellingham et al., 1 Nov 2025).

The parameter $\widetilde\alpha(G)$ provides a flexible and powerful lens through which to understand and unify extremal conditions for cycles, paths, and constructed obstacles in both bipartite and general graphs. It encapsulates the robust obstruction posed by large independent bipartitions, leading to sharp stability and extremal results across a range of Hamiltonicity-type phenomena.

Principal references: (Axenovich et al., 2020, Ehard et al., 2020, Kogan, 2020, McDiarmid et al., 2016, Li et al., 11 Jun 2025, Ellingham et al., 1 Nov 2025, Cheng et al., 20 Nov 2025)