On Specific Factors in Graphs

Abstract

It is well known that if $$G = (V, E)$$ G = ( V , E ) is a connected multigraph and $$X\subset V$$ X ⊂ V is a subset of even order, then G contains a spanning forest H such that each vertex from X has an odd degree in H and all the other vertices have an even degree in H. This spanning forest may have isolated vertices. If this is not allowed in H, then the situation is much more complicated. In this paper, we study this problem and generalize the concepts of even-factors and odd-factors in a unified form.

Sufficient conditions for graphs to have strong parity factors

“…Bujtás et al [2] introduced the concept of strong parity factor and conjectured that every 2-edge-connected graph with minimum degree at least three has a strong parity factor. Lu et al [13] showed a characterization for a graph to possess a strong parity factor and confirmed the above conjecture for 3-edge-connected graph.…”

“…A graph 𝐺 has a strong parity factor 𝐹 if for every subset 𝑋 ⊆ 𝑉 (𝐺) with |𝑋| even, 𝐺 contains a spanning subgraph 𝐹 such that 𝛿(𝐹 ) ≥ 1, 𝑑 𝐹 (𝑢) ≡ 1 (mod 2) for any 𝑢 ∈ 𝑋, and 𝑑 𝐹 (𝑣) ≡ 0 (mod 2) for any 𝑣 ∈ 𝑉 (𝐺)∖𝑋. Bujtás et al [2] introduced the concept of strong parity factor and derived some sufficient conditions for graphs with this property. Lu et al [13] presented a characterization for graphs to possess strong parity factors, which is shown in the following.…”

Modulo factors with bounded degrees

“…Remark 2.5. Note that Theorem 2.2 can conclude Theorem 11 in [4]. A generalization of it is formulated in [7,Theorem 6.5].…”

A characterization for graphs having strong parity factors

“…Bujtás, Jendrol, Tuza [3] introduced the definition of strong parity factor and gave some sufficient conditions for graphs to have this property. Especially, they showed that every 2-edge-connected graph with minimum degree three has the strong parity factor property and conjecture that the minimum degree condition can be improved.…”

“…Conjecture 1 (Bujtás, Jendrol and Tuza, [3]). Every 2-edge-connected graph of minimum degree at least three has the strong parity property.…”