Cycle Odd-Cover in Eulerian Graphs

• Cycle odd-cover is a mod-2 parity decomposition defined for Eulerian graphs, where a set of cycles has a symmetric difference equal to the graph's edge set.
• The framework introduces relaxed variants via topological and isolated vertex modifications, establishing sharp bounds based on maximum degree and linear arboricity.
• Distinct from cycle double covers and parity-constrained cycle-factors, cycle odd-cover emphasizes parity conditions over cycle lengths, clarifying its unique role in graph theory.

Cycle odd-cover is a parity-based cycle decomposition notion in graph theory. In the formulation introduced for Eulerian graphs, a cycle odd-cover of $G$ is a collection of cycles whose symmetric difference of edge sets is $E(G)$, and the minimum size of such a collection is denoted $c_2(G)$. The notion is part of a broader mod-$2$ covering framework that also includes path odd-covers and, in a distinct line of work, odd covers of cycle graphs by bicliques rather than by cycles (Borgwardt et al., 2023, Buchanan et al., 2022). A central theme across these formulations is that “odd” refers to parity of coverage under symmetric difference, not to the lengths of the covering cycles.

1. Definition, domain, and basic parameters

For an Eulerian graph $G$, a cycle odd-cover is a collection of cycles whose symmetric difference of edge sets is $E(G)$. Its minimum cardinality is denoted

$c_2(G).$

The cycle version is only defined for Eulerian graphs, because the class of Eulerian graphs is closed under symmetric difference (Borgwardt et al., 2023).

The same work introduces two relaxed variants. For a subdivision $H$ of $G$, $c_{2,\mathrm{top}}(G)$ is the minimum of $E(G)$0. For graphs obtained by adding isolated vertices, $E(G)$1 is the minimum of $E(G)$2. These are the cycle analogues of the topological and isolated-vertex relaxations defined first for path odd-covers (Borgwardt et al., 2023).

Parameter	Meaning	Scope
$E(G)$3	minimum size of a cycle odd-cover	Eulerian graphs
$E(G)$4	minimum $E(G)$5 over subdivisions $E(G)$6 of $E(G)$7	Eulerian graphs
$E(G)$8	minimum $E(G)$9 after adding isolated vertices	Eulerian graphs

This formulation is fundamentally mod-$c_2(G)$0: the covering object is not a multiset with prescribed exact multiplicities, but a family whose edge sets sum to $c_2(G)$1 over $c_2(G)$2. A plausible implication is that cycle odd-cover is best viewed as a parity-decomposition parameter rather than as a classical cover parameter in the exact-multiplicity sense.

2. Extremal bounds and exact statements for Eulerian graphs

The main cycle results currently recorded for this notion are three bounds stated for every Eulerian graph $c_2(G)$3: $c_2(G)$4

$c_2(G)$5

$c_2(G)$6

Here $c_2(G)$7 is the maximum degree and $c_2(G)$8 is the linear arboricity (Borgwardt et al., 2023).

The same source notes the immediate lower bound

$c_2(G)$9

and explicitly says that $2$0 is also a lower bound for $2$1 and $2$2 (Borgwardt et al., 2023).

The topological parameter is therefore pinned down exactly: $2$3 except in the special case where $2$4 is a union of two or more vertex-disjoint cycles, where the theorem does not claim equality (Borgwardt et al., 2023). This exactness is structurally notable because the ordinary parameter $2$5 is only given an upper bound in the same theorem, whereas the topological relaxation is completely resolved outside that exceptional family.

3. Transfer from path odd-covers and proof architecture

The cycle theory is presented as a transfer of the path odd-cover machinery. For paths, the odd-cover number $2$6 is the minimum cardinality of a collection of paths whose vertex sets are contained in $2$7 and whose symmetric difference of edge sets is $2$8. The analogous topological theorem for paths is

$2$9

for all graphs $G$0 except the trivial case of a disjoint union of at least one cycle with at most one path, while the general upper bound is

$G$1

(Borgwardt et al., 2023).

For cycles, the proof of

$G$2

transfers the path ideas by first decomposing an Eulerian graph into sets of vertex-disjoint cycles covering the maximum-degree vertices, then converting the path-based odd-cover constructions into cycle-based ones. Concretely, the argument uses repeated applications of the lemma that every Eulerian graph contains a set of vertex-disjoint cycles containing all maximum-degree vertices, and then each such set of cycles can be odd-covered by two paths; joining the two common endpoints of those paths produces two cycles. Iterating this over all cycle layers yields a cycle odd-cover with exactly $G$3 cycles (Borgwardt et al., 2023).

The cycle story is correspondingly cleaner than the path story. For paths, the odd-degree vertices and the maximum degree “compete” in the upper and lower bounds. For cycles, no such competition is needed because Eulerian graphs have $G$4, so the resulting statements are framed only in terms of $G$5 and $G$6 (Borgwardt et al., 2023).

4. Odd covers of cycle graphs by bicliques

A distinct literature uses “odd cover” in a biclique-decomposition sense. Given a finite simple graph $G$7, an odd cover of $G$8 is a collection of complete bipartite graphs, or bicliques, in which each edge of $G$9 appears in an odd number of bicliques and each non-edge of $E(G)$0 appears in an even number of bicliques. The minimum cardinality is denoted $E(G)$1 (Buchanan et al., 2022).

A central lower bound is

$E(G)$2

where

$E(G)$3

is the $E(G)$4-rank of the adjacency matrix. For cycle graphs, the paper gives exact values: $E(G)$5 and

$E(G)$6

(Buchanan et al., 2022).

Thus even and odd cycles behave differently. For even $E(G)$7, the rank lower bound is exact. For odd $E(G)$8, the exact value is one more than the rank lower bound. The upper bound construction for odd cycles uses $E(G)$9 copies of $c_2(G).$0 and one copy of $c_2(G).$1, while the lower bound is obtained by analyzing an arbitrary odd cover $c_2(G).$2, selecting a biclique $c_2(G).$3 with both $c_2(G).$4 and $c_2(G).$5 odd, and proving

$c_2(G).$6

which forces

$c_2(G).$7

(Buchanan et al., 2022).

This biclique-based theory is not the same object as cycle odd-cover in the sense of $c_2(G).$8, but it is directly relevant whenever the graph being covered is itself a cycle.

5. Disjoint unions of cycles and the even-core obstruction

The biclique odd-cover theory has also been pushed from single cycles to disjoint unions of cycles. If $c_2(G).$9 and $H$0, then

$H$1

Here the $H$2 are odd cycles, the $H$3 are even cycles, and $H$4 means disjoint union (Buchanan et al., 2024).

Several consequences are immediate. A single odd cycle satisfies

$H$5

while a single even cycle satisfies

$H$6

More generally, if a disjoint union of cycles contains at least one odd cycle, then its odd-cover number is exactly

$H$7

whereas a union of even cycles is rank-tight (Buchanan et al., 2024).

The decisive structural tool is the notion of an even core. A nonempty set $H$8 is an even core if every vertex of $H$9 has an even number of neighbors in $G$0. Equivalently, the rows of $G$1 indexed by $G$2 sum to $G$3 over $G$4. The paper proves that if $G$5 is a perfect odd cover, then every even core $G$6 must intersect both partite sets of every biclique in $G$7 evenly; in particular, if some induced subgraph on an even core has an odd number of edges, then $G$8 cannot have a perfect odd cover (Buchanan et al., 2024).

Applied to disjoint unions of cycles, an odd cycle component is itself an even core and induces an odd number of edges. That obstruction explains the global $G$9 correction term. This yields a sharp dichotomy: even-cycle components contribute exactly the rank bound, while the presence of any odd cycle forces one additional biclique (Buchanan et al., 2024).

6. Distinctions from cycle double covers and parity-constrained cycle-factors

Cycle odd-cover is often conflated with other parity-sensitive cycle notions, but the underlying requirements are different.

First, it is distinct from a cycle double cover. In a cycle double cover, every edge belongs to exactly two cycles. In the directed variant, every edge is covered exactly twice and the two covering cycles induce opposite orientations on that edge (Jiménez et al., 2014). By contrast, cycle odd-cover in the sense of $c_{2,\mathrm{top}}(G)$0 is a symmetric-difference condition on a family of cycles, defined only for Eulerian graphs (Borgwardt et al., 2023).

Second, it is distinct from approximate or embedding-based relaxations of the Cycle Double Cover conjecture. In that framework, a bridgeless cubic graph is studied through embeddings and singular edges, and the central equivalence is that the CDC conjecture is equivalent to bridgeless cubic graphs having an embedding with no singular edge (Ghanbari et al., 10 Nov 2025). This is a defect-minimization relaxation of exact double covering, not a mod-$c_{2,\mathrm{top}}(G)$1 odd-cover parameter.

Third, it is distinct from parity-constrained cycle-factors. A cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. The problem

$c_{2,\mathrm{top}}(G)$2

asks for a cycle-factor containing at least one odd cycle, whereas

$c_{2,\mathrm{top}}(G)$3

requires that all cycles be odd. These variants are NP-complete in directed, undirected, and mixed settings, except for the separate open status of the “at least one even cycle” problem in directed and undirected graphs (Hörsch et al., 21 Oct 2025).

These distinctions matter because the adjective “odd” can refer to three different features: odd multiplicity in a mod-$c_{2,\mathrm{top}}(G)$4 decomposition, odd cycle length inside a cycle-factor, or odd biclique coverage of the edge set of a graph. In the current graph-cover literature, cycle odd-cover most precisely refers to the Eulerian mod-$c_{2,\mathrm{top}}(G)$5 cycle decomposition measured by $c_{2,\mathrm{top}}(G)$6, while “odd covers of cycles” refers to the biclique parameter $c_{2,\mathrm{top}}(G)$7 and its extensions to disjoint unions of cycles (Borgwardt et al., 2023, Buchanan et al., 2024).