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A graph {\color{black}G} is Hamilton-connected if for any pair of distinct vertices {\color{black}u,v∈V(G)}, {\color{black}G} has a spanning (u,v)-path; {\color{black}G} is 1-hamiltonian if for any vertex subset S⊆V(G) with ∣S∣≤1, G−S has a spanning cycle. Let δ(G), α2˘7(G) and L(G) denote the minimum degree, the matching number and the line graph of a graph G, respectively. The following result is obtained. {\color{black} Let G be a simple graph} with ∣E(G)∣≥3. If δ(G)≥α2˘7(G), then each of the following holds. \\ (i) L(G) is Hamilton-connected if and only if κ(L(G))≥3. \\ (ii) L(G) is 1-hamiltonian if and only if κ(L(G))≥3. %==========sp For a graph G, an integer s≥0 and distinct vertices u,v∈V(G), an (s;u,v)-path-system of G is a subgraph H consisting of s internally disjoint (u,v)-paths. The spanning connectivity κ∗(G) is the largest integer s such that for any k with 0≤k≤s and for any u,v∈V(G) with uî€ =v, G has a spanning (k;u,v)-path-system. It is known that κ∗(G)≤κ(G), and determining if κ∗(G)3˘e0 is an NP-complete problem. A graph G is maximally spanning connected if κ∗(G)=κ(G). Let msc(G) and sk​(G) be the smallest integers m and m2˘7 such that Lm(G) is maximally spanning connected and κ∗(Lm2˘7(G))≥k, respectively. We show that every locally-connected line graph with connectivity at least 3 is maximally spanning connected, and that the spanning connectivity of a locally-connected line graph can be polynomially determined. As applications, we also determined best possible upper bounds for msc(G) and sk​(G), and characterized the extremal graphs reaching the upper bounds. %==============st For integers s≥0 and t≥0, a graph G is (s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G) with ∣X∣≤s and ∣Y∣≤t, G has a spanning closed trail that contains X and avoids Y. Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is (0,0)-supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29-45] showed that every simple graph G on n vertices with δ(G)≥5n​−1, when n is sufficiently large, is (0,0)-supereulerian or is contractible to K2,3​. We prove the following for any nonnegative integers s and t. \\ (i) For any real numbers a and b with 03˘ca3˘c1, there exists a family of finitely many graphs \F(a,b;s,t) such that if G is a simple graph on n vertices with κ2˘7(G)≥t+2 and δ(G)≥an+b, then either G is (s,t)-supereulerian, or G is contractible to a member in \F(a,b;s,t). \\ (ii) Let â„“K2​ denote the connected loopless graph with two vertices and â„“ parallel edges. If G is a simple graph on n vertices with κ2˘7(G)≥t+2 and δ(G)≥2n​−1, then when n is sufficiently large, either G is (s,t)-supereulerian, or for some integer j with t+2≤j≤s+t, G is contractible to a jK2​. %==================index For a hamiltonian property \cp, Clark and Wormold introduced the problem of investigating the value \cp(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: κ2˘7(G)≥a and δ(G)≥b}, and proposed a few problems to determine \cp(a,b) with b≥a≥4 when \cp is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let ess2˘7(G) denote the essential edge-connectivity of a graph G, and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: ess2˘7(G)≥a and δ(G)≥b}. We investigate the values of \cp\u27(a,b) when \cp is one of these hamiltonian properties. In particular, we show that for any values of b≥1, \cp\u27(4,b) \le 2 and \cp\u27(4,b) = 1 if and only if Thomassen\u27s conjecture that every 4-connected line graph is hamiltonian is valid
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