---
title: Odd Cycle Transversal Number
url: https://www.emergentmind.com/topics/odd-cycle-transversal-number
type: topic
---

# Odd Cycle Transversal Number

The odd cycle transversal number of a finite undirected graph $G=(V,E)$, denoted $\tau_{\mathrm{odd}}(G)$, is the minimum cardinality of a vertex subset $T\subseteq V$ such that every odd cycle $C\subseteq G$ has $T\cap V(C)\neq\emptyset$—that is, deleting $T$ from $G$ produces a bipartite graph. This parameter lies at the intersection of packing and covering theory and is a central object in the study of the Erdős–Pósa property for families of cycles, particularly odd cycles.

## 1. Formal Definition and Parameters

Given a graph $G=(V,E)$:
- The **odd cycle transversal number** $\tau_{\mathrm{odd}}(G)$ is defined as
  $$
  \tau_{\mathrm{odd}}(G) = \min\{\,|T|: T \subseteq V,\, T\cap V(C)\neq\emptyset\text{ for every odd cycle } C\subseteq G\,\}.
  $$
- The **odd cycle packing number** $\nu_{\mathrm{odd}}(G)$ is the largest $k$ such that $G$ contains $k$ vertex-disjoint odd cycles:
  $$
  \nu_{\mathrm{odd}}(G) = \max\{\,k : \exists\,C_1,\dots,C_k\text{ odd cycles, } V(C_i)\cap V(C_j)=\emptyset\ (i\neq j)\,\}.
  $$
- The **Erdős–Pósa ratio** for odd cycles is 
  $$
  R(G) = \frac{\tau_{\mathrm{odd}}(G)}{\nu_{\mathrm{odd}}(G)}
  $$
  with the convention $R(G)=\infty$ if $\nu_{\mathrm{odd}}(G)=0$ [2512.22865].

## 2. Erdős–Pósa Property and Known Results

The classic Erdős–Pósa property for a family $\mathcal{H}$ of subgraphs relates the maximum number of pairwise vertex-disjoint subgraphs from $\mathcal{H}$ to the minimum size of a vertex set intersecting all such subgraphs:
$$
\tau_\mathcal{H}(G) \leq f(\nu_\mathcal{H}(G))
$$
for some bounding function $f$ [1603.04615]. For all cycles, the optimal bounds are $\Theta(k\log k)$; however, for odd cycles, the property fails in general graphs. There exist graphs where $\nu_{\mathrm{odd}}(G)=1$ but $\tau_{\mathrm{odd}}(G)$ is arbitrarily large (e.g., high-girth bipartite blow-ups) [1603.04615].

Notably, for the subclass of **planar graphs**, the odd cycle family does exhibit the Erdős–Pósa property. Král', Sereni, and Stacho established $R(G)\leq 6$ for planar $G$; this was improved to $R(G)\leq 4$ [2512.22865]:

$$
\tau_{\mathrm{odd}}(G) \le 4\,\nu_{\mathrm{odd}}(G), \quad \text{for any planar } G
$$

## 3. Structural and Topological Techniques for Planar Graphs

The improvement to $R(G)\leq 4$ proceeds via the facial structure of planar embeddings:
- Construct the **vertex–face incidence graph** $VF(G)$ whose vertex set is $V(G)\cup \mathcal{F}(G)$ (the set of faces), and with edge $\{v, F\}$ iff $v$ lies on $F$.
- Hitting all odd cycles in $G$ is equivalent to hitting all odd faces in $VF(G)$ under a $T$-join–type condition.
- Odd faces are partitioned into **clouds** (maximal sets of odd faces connected in $VF(G)$), each handled separately.
- For clouds with odd-face packing number 1, any even subset can be hit by 2 vertices. For packing number $>1$, structural lemmata show that every even subset of certain subcollections can be hit by 4 vertices.
- Inductive merging and charging arguments, using planarity constraints in a reduced conflict graph, recursively bound the growth of the transversal during the process [2512.22865].

## 4. Tightness, Lower Bounds, and Examples

Tightness in the planar case remains open. The only known example achieving $R(G)=2$ is $K_4$ (one odd cycle, two vertices needed to hit). No graph is known with $R(G)>2$, so the gap $[2,4]$ is open. For cycles in general (not restricted to odd), the best possible is $\Theta(k\log k)$ [1603.04615, 2512.22865].

For general graphs (without planarity), the odd cycle transversal number can be unbounded with respect to the packing number—there is no Erdős–Pósa property for odd cycles [1603.04615].

## 5. Generalizations and Related Structures

Extensions and variants include:
- **Higher genus surfaces**: For fixed genus $g$, the odd-cycle Erdős–Pósa property holds with ratio $O(g)$; improving the proportionality constant remains open.
- **Edge transversals**: The edge version of the odd-cycle transversal has ratio exactly 2 in planar graphs.
- **Uncrossable families**: The methodology applies to uncrossable cycle families, yielding constant-factor ratios (best known is $3$ for cycles, $8.38$ for general uncrossable families).
- **Directed odd cycles**: In planar digraphs, directed odd-cycle transversals meet a constant-factor Erdős–Pósa ratio, but the constant is not known explicitly [2512.22865].

## 6. Algorithmic Implications and Open Problems

The constructive proof for $R(G)\leq 4$ in planar graphs yields a polynomial-time $4$-approximation for finding a minimum odd cycle transversal; the best-known ratio achieved by combinatorial algorithms is $2.4$. Whether a factor-2 approximation, matching the packing lower bound, can be achieved is open [2512.22865].

Other open problems include:
- Determining the exact supremum of $R(G)$ for planar graphs ($2\leq R(G)\leq 4$).
- Extension and sharpness for higher genus and other graph classes.
- The behavior of related transversals in the edge-setting and in more general host-classes.

## 7. Summary Table: Odd Cycle Transversal Bounds in Planar Graphs

| Parameter                       | Lower Bound | Upper Bound  | Reference      |
|----------------------------------|-------------|--------------|---------------|
| $\tau_{\mathrm{odd}}/\nu_{\mathrm{odd}}$ (planar) | 2 ($K_4$)  | 4            | [2512.22865]  |

## 8. Context Within the Erdős–Pósa Framework

The odd cycle transversal number is a canonical instantiation of the minimum hitting set for a guest-class of subgraphs (the odd cycles). It is exclusively for planar graphs that the odd cycle family enjoys the Erdős–Pósa property with a constant-ratio gap. This stands in marked contrast to other cyclic containment relations: all cycles ($\Theta(k\log k)$), induced cycles ($O(k^2\log k)$), long cycles ($O(k\ell + k\log k)$), and odd cycles (constant only for planar graphs) [1603.04615, 2512.22865, 1711.00667]. This dichotomy is sharp and topologically rooted. The determination of the precise constants and efficient algorithms for odd cycle transversals in planar and higher-genus graphs remains an active frontier.

Source: https://www.emergentmind.com/topics/odd-cycle-transversal-number