3 research outputs found
On the extremal number of edges in hamiltonian connected graphs
AbstractAssume that n and δ are positive integers with 3≤δ<n. Let hc(n,δ) be the minimum number of edges required to guarantee an n-vertex graph G with minimum degree δ(G)≥δ to be hamiltonian connected. Any n-vertex graph G with δ(G)≥δ is hamiltonian connected if |E(G)|≥hc(n,δ). We prove that hc(n,δ)=C(n−δ+1,2)+δ2−δ+1 if δ≤⌊n+3×(nmod2)6⌋+1, hc(n,δ)=C(n−⌊n2⌋+1,2)+⌊n2⌋2−⌊n2⌋+1 if ⌊n+3×(nmod2)6⌋+1<δ≤⌊n2⌋, and hc(n,δ)=⌈nδ2⌉ if δ>⌊n2⌋
Conditional fault hamiltonian connectivity of the complete graph
A path in G is a hamiltonian path if it contains all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path between any two distinct vertices of G. The degree of a vertex u in G is the number of vertices of G adjacent to u. We denote by is defined as the maximum integer k such that G is k edge-fault tolerant conditional hamiltonian connected if G is hamiltonian connected and is undefined otherwise. Let n 4. We use K n to denote the complete graph with n vertices. In this paper, we show tha
Integer k-matching preclusion of graphs
As a generalization of matching preclusion number of a graph, we provide the
(strong) integer -matching preclusion number, abbreviated as number
( number), which is the minimum number of edges (vertices and edges)
whose deletion results in a graph that has neither perfect integer -matching
nor almost perfect integer -matching. In this paper, we show that when
is even, the () number is equal to the (strong) fractional
matching preclusion number. We obtain a necessary condition of graphs with an
almost-perfect integer -matching and a relational expression between the
matching number and the integer -matching number of bipartite graphs. Thus
the number and the number of complete graphs, bipartite
graphs and arrangement graphs are obtained, respectively.Comment: 18 pages, 5 figure