It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U, of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either symbol + or ⊥. A multigraph on n vertices with exactly two traceable pairs is constructed for each n ≥ 3. The...

For n ≥ 4, the complete n-vertex multidigraph with arc multiplicity λ is proved to have a decomposition into directed paths of arbitrarily prescribed lengths ≤ n - 1 and different from n - 2, unless n = 5, λ = 1, and all lengths are to be n - 1 = 4. For λ = 1, a more general decomposition exists; namely, up to five paths of length n - 2 can also be prescribed.

The line graph of a graph $G$, denoted by $L\left(G\right)$, has $E\left(G\right)$ as its vertex set, where two vertices in $L\left(G\right)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$, define ${\overline{\sigma }}_2\left(H\right)=min\{d\left(u\right)+d\left(v\right):uv\in E\left(H\right)\}$. Let $H$ be a 2-connected claw-free simple graph of order $n$ with $\delta \left(H\right)\ge 3$. We show that, if ${\overline{\sigma }}_2\left(H\right)\ge \frac{1}{7}(2n-5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $\mathrm{cl}\left(H\right)=L\left(G\right)$, where $G$ is an essentially $2$-edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10 which have...

2010 Mathematics Subject Classification: 05C38, 05C45.In 1952, Dirac introduced the degree type condition and proved that if G is a connected graph of order n і 3 such that its minimum degree satisfies d(G) і n/2, then G is Hamiltonian. In this paper we investigate a further condition and prove that if G is a connected graph of order n і 3 such that d(G) і (n-2)/2, then G is Hamiltonian or G belongs to four classes of well-structured exceptional graphs.