Path Odd-Cover: Parity Graph Decomposition

• Path odd-cover is a parity-based graph decomposition notion that expresses a graph as the mod-2 sum of paths on its complete vertex set.
• It lies between classical path decompositions and linear arboricity, leveraging mod-2 cancellations to achieve tighter and more flexible coverings.
• Recent work refines degree-based upper bounds using matching and cycle pairing techniques, particularly improving results for Eulerian graphs.

Searching arXiv for recent and foundational papers on path odd-covers and closely related odd-cover notions. arxiv_search(query="path odd-covers of graphs odd covers graphs bicliques", max_results=10, sort_by="relevance") Searching arXiv for directly relevant papers. search_arxiv({"query":"path odd-covers of graphs odd covers graphs bicliques", "max_results": 10, "sort_by":"relevance"}) Path odd-cover is a parity-based graph decomposition notion in which a graph $G$ is represented as the symmetric difference of paths on the complete graph on $V(G)$. Formally, a path odd-cover of $G$ is a collection of paths $P_1,\dots,P_k$ such that

$$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$

where $\oplus$ denotes symmetric difference of edge sets; the minimum such $k$ is the path odd-cover number $p_2(G)$. The notion was introduced as a weakening of Gallai’s path decomposition problem and a strengthening of linear arboricity, and later work sharpened its degree-based upper bounds, especially for Eulerian graphs (Borgwardt et al., 2023, Borgwardt et al., 17 Jul 2025).

1. Formal definition and parity model

Let $G=(V,E)$ be a finite graph. A path odd-cover of $G$ is a collection of paths in the complete graph on $V(G)$0 whose mod-$V(G)$1 sum is exactly $V(G)$2. Equivalently, every edge of $V(G)$3 appears in an odd number of the chosen paths, and every nonedge of $V(G)$4 appears in an even number of them. The parameter $V(G)$5 is the minimum size of such a collection (Borgwardt et al., 2023).

A recurrent source of confusion is that the paths are not required to be subgraphs of $V(G)$6. They are paths on the vertex set $V(G)$7, so they may use nonedges of $V(G)$8, provided those extra edges cancel in the final symmetric difference. In the later general odd-cover framework, if the host graph is not specified it is taken to be the complete graph on $V(G)$9; a path odd-cover is then an odd-cover in which each summand is a path (Borgwardt et al., 17 Jul 2025).

This formulation is intrinsically $G$0-linear. The edge set of $G$1 is encoded as a mod-$G$2 sum of path edge-incidence vectors, so the problem belongs to the broader family of parity decomposition problems. A plausible implication is that structural progress is most naturally expressed through mod-$G$3 cancellations rather than through edge-disjoint decompositions.

2. Position among classical decomposition parameters

Path odd-cover sits between two classical parameters. If $G$4 denotes the path decomposition number and $G$5 the linear arboricity, then

$G$6

The upper bound holds because a path decomposition is a special case of a path odd-cover. The lower bound holds because from any path odd-cover with $G$7 paths, deleting duplicated edges in pairs yields a decomposition of $G$8 into $G$9 edge-disjoint linear forests (Borgwardt et al., 2023).

The parameter is therefore weaker than exact path decomposition and stronger than linear arboricity. This is the sense in which the original paper positions the notion between Gallai-type path decomposition problems and linear forest decompositions. The same paper also records the arboricity lower bound

$P_1,\dots,P_k$0

placing path odd-cover within a broader Hasse-diagram of graph covering invariants (Borgwardt et al., 2023).

Later work makes a second conceptual distinction explicit: every graph admits a path odd-cover, whereas a cycle odd-cover exists if and only if the graph is Eulerian (Borgwardt et al., 17 Jul 2025). This sharp separation explains why paths are the more flexible mod-$P_1,\dots,P_k$1 covering objects. The 2023 study also notes that the path covering number $P_1,\dots,P_k$2 is not well aligned with $P_1,\dots,P_k$3: the two can be arbitrarily far apart in either direction (Borgwardt et al., 2023).

3. General bounds and exact special cases

The first general bounds are degree- and parity-based. If $P_1,\dots,P_k$4 is the maximum degree and $P_1,\dots,P_k$5 is the number of odd-degree vertices, then

$P_1,\dots,P_k$6

and

$P_1,\dots,P_k$7

The lower bound is immediate because each path has at most two odd-degree endpoints and maximum degree $P_1,\dots,P_k$8, while the upper bound is the main theorem of the 2023 paper (Borgwardt et al., 2023).

The 2025 paper improves the upper bound. Writing $P_1,\dots,P_k$9 for the number of odd-degree vertices and $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$0, it proves that every graph $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$1 admits a path odd-cover of size at most

$$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$2

In particular, if $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$3 is Eulerian, then $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$4 admits a path odd-cover of size at most

$$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$5

improving the previous Eulerian upper bound $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$6 (Borgwardt et al., 17 Jul 2025).

Several exact evaluations clarify the parameter’s behavior. A path $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$7 satisfies $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$8, since one path suffices. A cycle $$E(G)=E(P_1)\oplus E(P_2)\oplus \cdots \oplus E(P_k),$$9 satisfies $\oplus$0, obtained by taking the two arcs between two chosen vertices. A star $\oplus$1 satisfies

$\oplus$2

The disjoint union of $\oplus$3 cycles satisfies $\oplus$4, while $\oplus$5, illustrating how strongly the use of nonedges and mod-$\oplus$6 cancellation can compress a decomposition (Borgwardt et al., 2023).

4. Topological and isolated-vertex relaxations

Two relaxations developed alongside $\oplus$7 are topological and isolated-vertex variants. The topological relaxation is

$\oplus$8

Subdivision preserves $\oplus$9 and $k$0, so the same lower bound remains available. The principal result is that if $k$1 is not the disjoint union of at least one cycle with at most one path, then

$k$2

In the exceptional case, the lower bound equals $k$3, but $k$4 (Borgwardt et al., 2023).

The isolated-vertex relaxation is

$k$5

For Eulerian graphs,

$k$6

Moreover, if $k$7, then

$k$8

This identifies a regime in which path odd-cover coincides exactly with linear arboricity after, or even without, augmentation by isolated vertices (Borgwardt et al., 2023).

The isolated-vertex relaxation is strictly weaker in general. For every odd integer $k$9, there exists an Eulerian graph $p_2(G)$0 with

$p_2(G)$1

This separation shows that the endpoint constraints of actual paths, even in a mod-$p_2(G)$2 setting, are not completely captured by linear forest structure alone (Borgwardt et al., 2023).

5. Proof architecture and constructive mechanisms

The 2023 upper-bound proof proceeds by reducing to the Eulerian case. The odd-degree vertices are paired by a matching $p_2(G)$3 in the complete graph on $p_2(G)$4, and one sets

$p_2(G)$5

Then $p_2(G)$6 is Eulerian. A key lemma shows that every Eulerian graph contains a set of vertex-disjoint cycles covering every vertex of maximum degree. Another lemma shows that a set of vertex-disjoint cycles can be odd-covered by two paths. The technical core is then a sequence of lemmas that integrate matching edges into those path covers while controlling exceptional configurations (Borgwardt et al., 2023).

The relaxation theorems use additional local operations. For subdivisions, the proof introduces a well-distributed path $p_2(G)$7-system and uses the operations $p_2(G)$8 and $p_2(G)$9 to reduce the number of path components while preserving the mod-$G=(V,E)$0 sum. For isolated vertices, the operation $G=(V,E)$1 merges path components inside a linear forest by adding one isolated vertex and adjusting two other forests. These mechanisms are the constructive reason the topological lower bound is usually tight and the isolated-vertex Eulerian parameter matches linear arboricity (Borgwardt et al., 2023).

The 2025 improvement has a different architecture. Its local base case proves that every Eulerian graph of maximum degree $G=(V,E)$2 admits a path odd-cover of size at most $G=(V,E)$3. The global Eulerian bound then decomposes a graph of maximum degree $G=(V,E)$4 into $G=(V,E)$5 edge-disjoint polycycles, pairs those polycycles, and applies the degree-$G=(V,E)$6 theorem to each pair; if one polycycle remains, it has a path odd-cover of size at most $G=(V,E)$7. This pairing mechanism yields $G=(V,E)$8 (Borgwardt et al., 17 Jul 2025).

The degree-$G=(V,E)$9 theorem itself refines the Akiyama–Exoo–Harary theorem on decomposing graphs of maximum degree $G$0 into three linear forests. The paper introduces endpoint-parity data

$G$1

and a common parity parameter $G$2. By constructing transversal matchings with odd intersection and then minimizing the total endpoint count, it forces

$G$3

so each of the three linear forests is in fact a single path. The paper does not state a formal algorithmic theorem with explicit complexity, but it describes the proofs as highly constructive and polynomial-time in principle (Borgwardt et al., 17 Jul 2025).

6. Related notions, terminological pitfalls, and open directions

Path odd-cover should be distinguished from other odd-cover notions in graph theory. In “Odd Covers of Graphs,” an odd cover of a graph means a family of bicliques, with minimum size $G$4. For path graphs,

$G$5

This differs fundamentally from path odd-cover in the sense above, where the same target graph satisfies $G$6. The two theories use different covering objects—bicliques versus paths—and answer different mod-$G$7 representation questions (Buchanan et al., 2022).

It should also be distinguished from the packing-and-covering theory of odd $G$8-trails. There the objects are odd trails, covers are edge sets intersecting every such trail, and the central theorem is

$G$9

with this bound tight. That theory is explicitly about trails rather than simple paths, and the paper emphasizes that the analogous odd-path packing/covering gap is unbounded. A plausible implication is that path odd-cover gains tractability not from a min-max theorem of Menger type, but from allowing global mod-$V(G)$00 cancellation among path summands (Ibrahimpur et al., 2017).

Several open problems remain central. The 2023 paper asks whether there exists a graph $V(G)$01 with

$V(G)$02

and conjectures that every $V(G)$03-vertex graph satisfies

$V(G)$04

The 2025 work resolves the earlier degree-$V(G)$05 Eulerian question positively, but it proposes a stronger asymptotic conjecture: there exists a constant $V(G)$06 such that every Eulerian graph $V(G)$07 admits a path odd-cover of size $V(G)$08 (Borgwardt et al., 2023, Borgwardt et al., 17 Jul 2025). This suggests that the true growth rate may lie closer to $V(G)$09 than to the current $V(G)$10 upper bound.