Odd Cycle Transversal Number

• Odd Cycle Transversal Number is a graph parameter defined as the minimum vertex set needed to intersect every odd cycle, ensuring the remaining graph is bipartite.
• It is central to the study of the Erdős–Pósa property for odd cycles, linking the concepts of packing and covering in graph theory.
• In planar graphs, structural techniques yield constant-factor bounds between 2 and 4, leading to efficient approximation algorithms.

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 $\nu_{\mathrm{odd}}(G)=0$ (Puhlmann et al., 28 Dec 2025).

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 $f$ (Raymond et al., 2016). 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) (Raymond et al., 2016).

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 $R(G)\leq 4$ (Puhlmann et al., 28 Dec 2025):

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

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 (Puhlmann et al., 28 Dec 2025).

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 $\Theta(k\log k)$ (Raymond et al., 2016, Puhlmann et al., 28 Dec 2025).

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 (Raymond et al., 2016).

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 (Puhlmann et al., 28 Dec 2025).

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 (Puhlmann et al., 28 Dec 2025).

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	(Puhlmann et al., 28 Dec 2025)

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) (Raymond et al., 2016, Puhlmann et al., 28 Dec 2025, Kim et al., 2017). 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.