Spanning trees and spanning Eulerian subgraphs with small degrees

Abstract

Let G be a graph with X ⊆ V (G) and let l be an intersecting supermodular subadditive integer-valued function on subsets of V (G). The graph G is said to be l-partition-connected, if for every partition P of, where eG(P ) denotes the number of edges of G joining different parts of P . Let λ ∈ [0, 1] be a real number and let η be a real function on X. In this paper, we show that if G is l-partition-connected and for all S ⊆ X,then G has an l-partition-connected spanning subgraph H such that for each vertex v ∈ X, dH (v) ≤ ⌈η(v) − λl(v)⌉, where eG(S) denotes the number of edges of G with both ends in S and Θ l (G \ S) denotes the maximum number of all A∈P l(A) − e G\S (P ) taken over all partitions P of V (G) \ S. Finally, we show that if H is an (l1 + · · · + lm)-partition-connected graph, then it can be decomposed into m edge-disjoint spanning subgraphs H1, . . . , Hm such that every graph Hi is li-partition-connected, where l1, l2, . . . , lm are m intersecting supermodular subadditive integer-valued functions on subsets of V (H).These results generalize several known results.

Highly edge‐connected factors using given lists on degrees

“…■ Theorem 4 directly gives the following corollary. This statement was already shown in [9], and implicitly in [4] for = 2, 6. Corollary 1.…”

“…Theorem directly gives the following corollary. This statement was already shown in , and implicitly in for . Corollary Every 2 k ‐edge‐connected graph G has a spanning k ‐tree‐connected subgraph H such that for every , .…”

“…Note that the 3‐edge‐connectedness in 5‐regular graphs guarantees the existence of a connected ‐factor ([, Corollary ]), but the 2‐edge‐connectedness cannot (see Figure ). Theorem says that the 2‐edge‐connectedness for 5‐regular graphs guarantees the existence of a connected ‐factor.…”

Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

“…Apart from many edge-disjoint spanning trees, large edge-connectivity also guarantees the existence of spanning trees with small vertex degrees. This has been investigated by several authors, see for example [6], [8], [10], and [24]. Small-degree spanning trees have already been used in [1], [18], [19], and [22] to prove special cases of Conjecture 1.1.…”

On Specific Factors in Graphs

“…Connected factors, especially spanning trees, of specific properties have been extensively studied as well; see e.g. Chapter 8 in [2] and surveys in the papers [6,9], and [12]. From that area we will employ the following result of Thomassen [12].…”