Note on parity factors of regular graphs

In this paper, we obtain a sufficient condition for the existence of parity factors in a regular graph in terms of edge-connectivity. Moreover, we also show that our condition is sharp.


Preliminaries
Let G = (V, E) be a graph with vertex set V (G) and edge set E(G). The number of vertices of a graph G is called the order of G and is denoted by n. On the other hand, the number of edges of G is called the size of G and is denoted by e. For a vertex v of graph G, the number of edges of G incident with v is called the degree of v in G and is denoted by d G (v). For two subsets S, T ⊆ V (G), let e G (S, T ) denote the number of edges of G joining S to T .
Let therefore g, f : V → Z + such that g(v) ≤ f (v) and g(v) ≡ f (v) (mod 2) for every v ∈ V . Then a spanning subgraph F of G is called a (g, f )-parity-factor, if Let a, b be two integers such that 1 ≤ a ≤ b and a ≡ b (mod 2). If g(v) ≡ a and f (v) ≡ b for all v ∈ V (G), then a (g, f )-parity-factor is called (a, b)-parity factor. When a = 1, (a, b)-parity factor is called (1, n)-odd factor.
For a general graph G and an integer k, a spanning subgraph F such that is called a k-factor. In fact, a k-factor is also a (k, k)-parity factor. Now let us recall one of the most classic results due to Petersen. Theorem 1.1 (Petersen [8]) Let r and k be integers such that 1 ≤ k ≤ r. Every 2rregular graph has a 2k-factor.
By the edge-connectivity, Gallai [4] proved the following result. * This work was supported by the National Natural Science Foundation of China (No. 11101329) † Corresponding email: luhongliang215@sina.com (H. Lu) Theorem 1.2 (Gallai [4]) Let r and k be integers such that 1 ≤ k < r, and G an m-edgeconnected r-regular graph, where m ≥ 1. If one of the following conditions holds, then G has a k-factor.
(i) r is even, k is odd, |G| is even, and r m ≤ k ≤ r(1 − 1 m ); (ii) r is odd, k is even and 2 ≤ k ≤ r(1 − 1 m ); (iii) r and k are both odd and r m ≤ k.

Theorem 1.3 (Bollobás, Saito and Wormald )
Let r and k be integers such that 1 ≤ k < r, and G be an m-edge-connected r-regular graph, where m ≥ 1 is a positive integer. Let m * ∈ {m, m + 1} such that m * ≡ 1 (mod 2). If one of the the following conditions holds, then G has a k-factor.
(i) r is odd, k is even and 2 ≤ k ≤ r(1 − 1 m * ); (ii) r and k are both odd and r m * ≤ k.
In this paper, we extend Gallai as well as Bollobás, Satio and Wormald result to (a, b)parity factor. The main tool in our proofs is the famous theorem of Lovász (see [7]).

Main Theorem
Theorem 2.1 Let a, b and r be integers such that 1 ≤ a ≤ b < r and a ≡ b (mod 2). Let G be a m-edge-connected r-regular graph with n vertices. If one of the following conditions holds, then G has a (a, b)-parity factor.
(i) r is even, a, b are odd, |G| is even, r m ≤ b and a ≤ r(1 − 1 m ); (ii) r is odd, a, b are even and a ≤ r(1 − 1 m * ); (iii) r, a, b are odd and r m * ≤ b.
Proof. By Theorem 1.3, (ii) and (iii) are followed. Now we prove (i). Let θ 1 = a r and θ 2 = b r . Then 0 < θ 1 ≤ θ 2 < 1. Suppose that G contains no (a, b)-parity factors. By Theorem 1.4, there exist two disjoint subsets S and T of V (G) such that S ∪ T = ∅, and where τ is the number of a-odd (i.e. b-odd) components C of G − (S ∪ T ). Let C 1 , · · · , C τ denote a-odd components of G − S − T and D = C 1 ∪ · · · ∪ C τ .
Note that Since G is connected and 0 < θ 1 ≤ θ 2 < 1, so θ 2 e G (S, C i ) + (1 − θ 1 )e G (T, C i ) > 0 for each C i . Hence we will obtain a contradiction by showing that for every C = C i , 1 ≤ i ≤ τ , we have These inequalities together with the previous inequality imply which is impossible. Since C is a a-odd component of G − (S ∪ T ), we have a|C| + e G (T, C) ≡ 1 (mod 2).
Moreover, since r|C| = It is obvious that the two inequalities e G (S, C) ≥ 1 and e G (T, C) ≥ 1 imply Hence we may assume e G (S, C) = 0 or e G (T, C) = 0.  (2), we have If e G (T, C) = 0, then e G (S, C) ≥ m. Since r m ≤ b, hence θ 2 m ≥ 1, and so we obtain Consequently, condition (i) guarantees (2) holds and thus (i) is true. Consequently the proof is complete. ✷

Remark:
The edge-connectivity conditions in Theorem 2.1are sharp.
We give the description for (i). For (ii) and (iii), the constructions are similar but slightly more complicated. Let r ≥ 2 be an even integer, a, b ≥ 1 two odd integers and 2 ≤ m ≤ r−2 an even integer such that b < r/m or r(1 − 1 m ) < a. Since G has a (a, b)-parity factor if and only if G has a (r − b, r − a)-parity factor, so we can assume b < r/m. Let J(r, m) be the complete graph K r+1 from which a matching of size m/2 is deleted. Take r disjoint copies of J(r, m). Add m new vertices and connect each of these vertices to a vertex of degree r − 1 of J(r, m). This gives an m-edge-connected r-regular graph denoted by G. Let S denote the set of m new vertices and T = ∅. Let τ denote the number of components C, called a-odd components of G − (S ∪ T ) such that e G (V (C), T ) + a|C| ≡ 1 (mod 2). Then we have τ = r, and