A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let κ2˘7(G) and δ(G) be the edge-connectivity and the minimum degree of a graph G, respectively. 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. This dissertation is devoted to providing some results on (s,t)-supereulerian graphs and supereulerian hypergraphs.
In Chapter 2, we determine the value of the smallest integer j(s,t) such that every j(s,t)-edge-connected graph is (s,t)-supereulerian as follows:
j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right.
As applications, we characterize (s,t)-supereulerian graphs when t≥3 in terms of edge-connectivities, and show that when t≥3, (s,t)-supereulerianicity is polynomially determinable.
In Chapter 3, for a subset Y⊆E(G) with ∣Y∣≤κ2˘7(G)−1, a necessary and sufficient condition for G−Y to be a contractible configuration for supereulerianicity is obtained. We also characterize the (s,t)-supereulerianicity of G when s+t≤κ2˘7(G). These results are applied to show that if G is (s,t)-supereulerian with κ2˘7(G)=δ(G)≥3, then for any permutation α on the vertex set V(G), the permutation graph α(G) is (s,t)-supereulerian if and only if s+t≤κ2˘7(G).
For a non-negative integer s≤∣V(G)∣−3, a graph G is s-Hamiltonian if the removal of any k≤s vertices results in a Hamiltonian graph. Let is,t(G) and hs(G) denote the smallest integer i such that the iterated line graph Li(G) is (s,t)-supereulerian and s-Hamiltonian, respectively. In Chapter 4, for a simple graph G, we establish upper bounds for is,t(G) and hs(G). Specifically, the upper bound for the s-Hamiltonian index hs(G) sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
Harary and Nash-Williams in 1968 proved that the line graph of a graph G is Hamiltonian if and only if G has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity.
Applying the adjacency matrix of a hypergraph H defined by Rodr\\u27iguez in 2002, let λ2(H) be the second largest adjacency eigenvalue of H. In Chapter 6, we prove that for an integer k and a r-uniform hypergraph H of order n with r≥4 even, the minimum degree δ≥k≥2 and k=r+2, if λ2(H)≤(r−1)δ−4(r+1)(n−r−1)r2(k−1)n, then H is k-edge-connected. %κ2˘7(H)≥k.
Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The (s,t)-supereulerianicity of hypergraphs is another interesting topic to be investigated in the future