Factorizing regular graphs

Abstract

Every 9-regular graph (possibly with multiple edges) with odd edge-connectivity > 5 can be edge-decomposed into three 3-factors. If Tutte's 3-flow conjecture is true, it also holds for all 9-regular graphs with odd edge-connectivity 5, but not with odd edge-connectivity 3. It holds for all planar 2-edge-connected 9-regular graphs, an equivalent version of the 4-color theorem for planar graphs. We address the more general question: If G is an r-regular graph, and r = kq where k, q are natural numbers > 1, can G be edge-decomposed into k q-factors? If q is even, then the decomposition exists trivially. If k, q are both odd, then we prove that the decomposition exists if G has odd edgeconnectivity (size of smallest odd edge-cut) at least 3k − 2, which is satisfied if the odd edge-connectivity is at least r − 2. If q is odd and k is even, then we must require that G has an even number of vertices just to guarantee that G has a q-factor. If we want a decomposition into q-factors, then we also need the condition that, for any partition of the vertex set of G into two odd parts, there must be at least k edges between the parts. We prove that the edge-decomposition into q-factors is always possible if G has an even number of vertices and the edge-connectivity of G is at least 2k 2 + k.

Equitable factorizations of edge-connected graphs

“…The following corollary gives a supplement for Theorem 1.4 and partially confirms Conjecture 2 in [20].…”

“…In this section, we shall formulate following theorem on existence of equitable factorizations in directed general graphs that provides a relationship between orientations and factorizations of general graphs and is motivated by Theorem 2 in [20].…”

“…Proof. The proof presented here is inspired by the proof of Theorem 2 in [20]. First split every vertex We claim that for any two colors c i and c j , we have |m i − m j | ≤ 1.…”

“…Theorem 1.4. ( [20]) Let k and r be two odd positive integers with r ≥ 3. If G is an odd-(3k − 2)-edgeconnected kr-regular general graph, then it can be edge-decomposed into r-regular factors.…”

Highly edge‐connected regular graphs without large factorizable subgraphs

“…For , a graph is a ‐ factor of , if is a spanning ‐regular subgraph of . Recently, Thomassen stated the following problem (see Problem 1 in [4]).…”