4,791 research outputs found
A generalization of Ore's Theorem involving neighborhood unions
AbstractLet G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-connected and |N(u)∪N(v)|⩾12n for every pair of nonadjacent vertices u,v, then either G is hamiltonian, or G is the Petersen graph, or G belongs to one of three families of exceptional graphs of connectivity 2
Existence of Dλ-cycles and Dλ-paths
A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way
On factors of 4-connected claw-free graphs
We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is Hamiltonian, i.e. has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass-free graphs, i.e. graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjecture 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths. \u
On the minimum leaf number of cubic graphs
The \emph{minimum leaf number} of a connected graph is
defined as the minimum number of leaves of the spanning trees of . We
present new results concerning the minimum leaf number of cubic graphs: we show
that if is a connected cubic graph of order , then , improving on the best known result in [Inf. Process.
Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph
Theory and Applications 5 (2017) 207-211]. We further prove that if is also
2-connected, then , improving on the best
known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new
conjectures concerning the minimum leaf number of several types of cubic graphs
and examples showing that the bounds of the conjectures are best possible.Comment: 17 page
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