We show that if a 2-edge connected graph G has a unique f-factor F, then some vertex has the same degree in F as in G. This conclusion is the best possible, even if the hypothesis is considerably strengthened.
A theorem of J. Edmonds states that a directed graph has k edge-disjoint branchings rooted at a vertex r if and only if every vertex has k edge-disjoint paths to r . We conjecture an extension of this theorem to vertex-disjoint paths and give a constructive proof of the conjecture in the case k = 2.
THE CONJECTURELet G = ( V , E ) be a finite directed graph with vertex set V and edge set E . Multiple edges are allowed, but self loops are excluded. An edge directed from x to y will be denoted by (x,y) (we do not distinguish between multiple edges from x to y), and we refer to x as the tail and y as the head of this edge. If P is a directed path from x to y , we refer to x and y as the tail and head of P , respectively, and say that P is trivial if x and y are the same vertex. If x' and y ' are the tail and head of a subpath of P , we write P :x' + y ' for the restriction of P to this path. Two paths are called edge-disjoint if they have no common edge and vertex-disjoint if they have no common vertices except possibly a common head or tail (the trivial path x is vertex-disjoint from precisely those paths with head or tail x). For a subset R of V , an R-branching in G is a spanning forest B of G in which all vertices of V -R have outdegree precisely 1. When R just consists of a single vertex r, we refer to B as an r-branching.Let I = (1, . . . , k}, where 1 5 k 5 IVI. Let R = { R, 1 i E I } be a family of subsets of V, and let B = {B, I i E I } be a family of edge-disjoint branchings in G such that B, is an R,-branching. Let X = {x, I i E I } be a family of (possibly
It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly 'intermediate' variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords? For fixed k, we establish a lower bound of Ω n 1/k on the growth rate.
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