Nash-Williams and Tutte independently characterized when a graph has k edge-disjoint spanning trees; a consequence is that 2k-edge-connected graphs have k edge-disjoint spanning trees. Kriesell conjectured a more general statement: defining a set S ⊆ V (G) to be j-edge-connected in G if S lies in a single component of any graph obtained by deleting fewer… (More)
It is shown that every (2p + 1) log 2 (|V (G)|)-edge-connected graph G has a mod (2p + 1)-orientation, and that a (4p + 1)-regular graph G has a mod (2p + 1)-orientation if and only if V (G) has a partition (V + , V −) such that ∀U ⊆ V (G), These extend former results by Da Silva and Dahad on nowhere zero 3-flows of 5-regular graphs, and by Lai and Zhang on… (More)
The Chvátal–Erd˝ os Theorem states that every graph whose connectivity is at least its independence number has a spanning cycle. In 1976, Fouquet and Jolivet conjectured an extension: If G is an n-vertex k-connected graph with independence number a, and a ≥ k, then G has a cycle with length at least k(n+a−k) a. We prove this conjecture.
A hub set in a graph G is a set U ⊆ V (G) such that any two vertices outside U are connected by a path whose internal vertices lie in U. We prove that h(G) ≤ h c (G) ≤ γ c (G) ≤ h(G) + 1, where h(G), h c (G), and γ c (G), respectively, are the minimum sizes of a hub set in G, a hub set inducing a connected subgraph, and a connected dominating set.… (More)
Thomassen conjectured that every 4-connected line graph is Hamiltonian. A vertex cut X of G is essential if G − X has at least two non-trivial components. We prove that every 3-connected, essentially 11-connected line graph is Hamiltonian. Using Ryjá˘ cek's line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is… (More)
A parity walk in an edge-coloring of a graph is a walk traversing each color an even number of times. We introduce two parameters. Let p(G) be the least number of colors in a parity edge-coloring of G (a coloring having no parity path). Let b p(G) be the least number of colors in a strong parity edge-coloring of G (a coloring having no open parity walk).… (More)
We prove a conjecture of Ohba which says that every graph G on at most 2χ(G) + 1 vertices satisfies χ ℓ (G) = χ(G).
The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ |V (G)|, Hadwiger's Conjecture implies that α(G)h(G) ≥ |V (G)|. We show that (2α(G) − ⌈log τ (τ α(G)/2)⌉)h(G) ≥ |V (G)| where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of… (More)
Say that a graph with maximum degree at most d is d-bounded. For d > k, we prove a sharp sparseness condition for decomposability into k forests and a d-bounded graph. Consequences are that every graph with fractional arboricity at most k + d k+d+1 has such a decomposition, and (for k = 1) every graph with maximum average degree less than 2 + 2d d+2… (More)
A graph is claw-free if it does not have an induced subgraph isomorphic to a K 1,3. In this paper, we proved the every 3-connected, essentially 11-connected claw-free graph is hamiltonian. We also present two related results concerning hamiltonian claw-free graphs.