Odd Cycle Packing in Graphs

• Odd cycle packing number is defined as the maximum count of vertex-disjoint odd cycles in a graph, serving as a measure of non-bipartiteness.
• It underpins algorithmic results and integer programming formulations, with known bounds such as the Erdős–Pósa ratio and polynomial-time methods in specific graph classes.
• Structural insights link it to chromatic numbers, stable set polytopes, and grid theorems, with current research addressing PTAS and tighter approximation bounds.

The odd cycle packing number, denoted $\nu_{\mathrm{odd}}(G)$ or $\ocp(G)$, quantifies the maximum size of a collection of pairwise vertex-disjoint odd cycles in an undirected graph $G=(V,E)$. As a fundamental invariant of non-bipartiteness, the odd cycle packing number is pivotal for structural graph theory, integer programming formulations, and approximation algorithms. Its connections to chromatic number, stable set polytopes, and the celebrated Erdős–Pósa property underpin deep algorithmic and extremal phenomena in both planar and topologically embedded graphs.

1. Formal Definitions and Variants

Let $G=(V,E)$ be a finite undirected graph.

• The odd cycle packing number $\nu_{\mathrm{odd}}(G) = \ocp(G)$ is defined as:

$$\nu_{\mathrm{odd}}(G) := \max \left\{ k : \text{there exist } k \text{ vertex-disjoint odd cycles in } G \right\}$$

Each cycle $C_i$ must be of odd length and the cycles must be vertex-disjoint.

• The odd cycle transversal number $\tau_{\mathrm{odd}}(G)$ is:

$$\tau_{\mathrm{odd}}(G) := \min \left\{ |T| : T \subseteq V, ~T \cap V(C) \neq \emptyset \text{ for every odd cycle } C \text{ in } G \right\}$$

That is, $\tau_{\mathrm{odd}}(G)$ is the smallest set of vertices intersecting every odd cycle.

• The induced odd cycle packing number $\iocp(G)$ restricts attention to induced odd cycles, where for each pair of cycles there are no edges between them outside the cycles themselves (Dvořák et al., 2020).

$\iocp(G) := \max \{ k : G \text{ has } k \text{ pairwise vertex-disjoint induced odd cycles} \}$

It holds that $\iocp(G) \leq \ocp(G)$. The parameter $\ocp(G)$ relates to determinants: if $M$ is the edge-node incidence matrix of $G$, then

$\max \{ |\det A| : A \text{ is a square submatrix of } M \} = 2^{\ocp(G)}$

(Conforti et al., 2019).

2. The Erdős–Pósa Property and Ratio

The Erdős–Pósa property asserts the existence of a function $f$ such that for a given cycle family $\mathcal{C}$ in a class of graphs $\mathcal{G}$,

$$\tau(G) \leq f(\nu(G)), \quad \forall G \in \mathcal{G}$$

For odd cycles, this property holds in restricted classes (including planar graphs and bounded-genus embeddings) and fails in general. The Erdős–Pósa ratio for odd cycles is:

$$\rho(G) := \frac{ \tau_{\mathrm{odd}}(G) }{ \nu_{\mathrm{odd}}(G) }$$

A class $\mathcal{C}$ has the quantitative Erdős–Pósa property for odd cycles if $\rho(G) \leq c$ for all $G\in\mathcal{C}$ and some constant $c$.

The best known bound in general planar graphs is $\rho(G) \leq 4$ (Puhlmann et al., 28 Dec 2025), improving the previous best $\rho(G)\leq 6$ by Král’, Sereni, and Stacho. In more general settings, the integrality gap and ratio admit upper bounds $<8.38$ via LP relaxations and rounding (Schlomberg, 2024).

3. Algorithmic Consequences and Polynomial-Time Results

In planar and bounded-genus graphs, several algorithmic ramifications follow:

• There exists a polynomial-time algorithm that, given $G$ planar, outputs a packing of odd cycles and a hitting set $T$ with $|T| \leq 4 \cdot |\mathcal{P}|$ (Puhlmann et al., 28 Dec 2025).
• For any uncrossable family of cycles (including odd cycles) in a planar $G$, one can round the optimal packing LP to an integral packing with value at least $1/\beta$, $\beta=(20+\sqrt{130})/9<3.5$, yielding

$$\nu_{\mathrm{odd}}(G) \geq \frac{1}{\beta} \cdot \mathrm{LP_{opt}}$$

and combined with dual rounding delivers $$\tau_{\mathrm{odd}}(G) < 8.38\,\nu_{\mathrm{odd}}(G)$$ (Schlomberg, 2024).

• In graphs of bounded odd-cycle-packing number and bounded genus, the stable set problem (and weighted variants) are solvable in polynomial time by reduction to circulation in a prescribed homology class of the dual graph (Conforti et al., 2019). All steps admit polynomial-size extended formulations for the stable-set polytope.
• Bounded induced odd-cycle packing number $\iocp(G)\le k$ suffices for randomized EPTAS and QPTAS for the independence number, without VC dimension assumptions (Dvořák et al., 2020).

4. Structural Theorems: Odd-Cycle-Packing-Treewidth and Grid Theorem

The odd-cycle-packing-treewidth (OCP-tw), denoted $\ocptw(G)$, measures the complexity of decomposing $G$ into bags with controlled odd-cycle-packing number. Formally, an OCP-tree-decomposition consists of a tree-decomposition $(T,\beta)$ augmented by apex sets $\alpha(t)\subseteq\beta(t)$ such that in each bag, removing $\alpha(t)$ yields at most $\ocp(G[\beta(t)\setminus\alpha(t)])$ vertex-disjoint odd cycles.

Analogous to the Grid Theorem, the presence of special grid-like minors (parity handles $H_k$ or parity vortices $V_k$ with $\ocp(H_k)=\ocp(V_k)=k$) certifies large OCP-tw. In their absence, $\ocptw(G)$ is bounded by a polynomial function of $k$, and an OCP-tree-decomposition can be constructed in FPT time $2^{\mathrm{poly}(k)}n^{O(1)}$ (Choi et al., 13 Nov 2025).

This structure enables polynomial-time algorithms for weighted Maximum Independent Set in graphs of bounded $\ocp(G)$.

5. Extremal Examples, Tightness, and Open Problems

For planar graphs, the known lower bound is $\rho(G) \geq 2$ (e.g., $G=K_4$ yields $\nu_{\mathrm{odd}}(K_4)=1$, $\tau_{\mathrm{odd}}(K_4)=2$), while no planar example with $\rho(G)>2$ is known (Puhlmann et al., 28 Dec 2025). Whether the true supremum in planar graphs is exactly $2$ or strictly between $2$ and $4$ remains unresolved.

For graphs of bounded genus, similar Erdős–Pósa type results hold for 2-sided odd cycles, with hitting set size $f(g,k)=19^{g+1}\cdot k$ in orientable surfaces (Conforti et al., 2019). The function $f$ and potential combinatorial optimizations are active open questions.

The existence of a true PTAS (rather than QPTAS) for independence number in classes $\iocp(G)\leq k$ is unsettled, as is the minimum-cost circulation algorithm for growing genus $g$ in bounded $\ocp(G)$ graphs.

6. Interactions with Chromatic Number, Stable Sets, and Integer Programming

Bounded (induced) odd cycle packing number has structural consequences for $\chi$-boundedness. For triangle-free graphs with $\iocp(G)\leq k$, one obtains $\chi(G)\leq 2+5k$, tightening as girth increases (Dvořák et al., 2020). More generally, for fixed $k$, $\chi(G)\leq f(\iocp(G),\omega(G)) = O(\omega(G)^{3k+O(1)})$.

In integer programming, the odd cycle packing number controls the largest subdeterminant in the edge-node incidence matrix and enables tractable algorithms for classes of integer programs where the constraint matrices admit tree-decompositions into totally delta-modular matrices with two non-zero entries per row (Choi et al., 13 Nov 2025).

7. Generalizations and Future Directions

Techniques for odd cycle packing and transversal extend to uncrossable cycle families, with LP-ratio bounds in planar graphs immediately applicable to more general cut- and flow-type problems (Schlomberg, 2024).

Research directions include:

• Improving the bound for $\rho(G)$ in planar and higher-genus graphs.
• Tight $\chi$-bounding functions for graphs with bounded induced or non-induced odd cycle packing number.
• Extension of polynomial-time algorithms to non-planar cases.
• Circulation algorithms with explicit bounds for large genus and efficient extended formulations.

The odd cycle packing number remains a central measure in graph theory, linking non-bipartiteness, topological constraints, and algorithmic tractability across a range of combinatorial and polyhedral frameworks.