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⌋

On the spanning connectivity and spanning laceability of hypercube-like networks

AbstractLet u and v be any two distinct nodes of an undirected graph G, which is k-connected. For 1≤w≤k, a w-container C(u,v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A w-container C(u,v) of G is a w∗-container if it contains all the nodes of G. A graph G is w∗-connected if there exists a w∗-container between any two distinct nodes. A bipartite graph G is w∗-laceable if there exists a w∗-container between any two nodes from different parts of G. Let G0=(V0,E0) and G1=(V1,E1) be two disjoint graphs with |V0|=|V1|. Let E={(v,ϕ(v))∣v∈V0,ϕ(v)∈V1, and ϕ:V0→V1 is a bijection}. Let G=G0⊕G1=(V0∪V1,E0∪E1∪E). The set of n-dimensional hypercube-like graph Hn′ is defined recursively as (a) H1′={K2}, K2= complete graph with two nodes, and (b) if G0 and G1 are in Hn′, then G=G0⊕G1 is in Hn+1′. Let Bn′={G∈Hn′ and G is bipartite} and Nn′=Hn′∖Bn′. In this paper, we show that every graph in Bn′ is w∗-laceable for every w, 1≤w≤n. It is shown that a constructed Nn′-graph H can not be 4∗-connected. In addition, we show that every graph in Nn′ is w∗-connected for every w, 1≤w≤3

Solution to an open problem on 4-ordered Hamiltonian graphs

AbstractA graph G is k-ordered if for any sequence of k distinct vertices of G, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs was posed in Ng and Schultz (1997) [10]. At the time, the only known examples were K4 and K3,3. Some progress was made in Mészáros (2008) [9] when the Peterson graph was found to be 4-ordered and the Heawood graph was proved to be 4-ordered Hamiltonian; moreover an infinite class of 3-regular 4-ordered graphs was found. In this paper we show that a subclass of generalized Petersen graphs are 4-ordered and give a complete classification for which of these graphs are 4-ordered Hamiltonian. In particular, this answers the open question regarding the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs. Moreover, a number of results related to other open problems are presented