Embedding Spanning Graphs In The Pseudorandom Graphs

Embedding subgraph is a classic problem in graph theory.In this paper,we mainly focus on embedding spanning subgraph in the pseudorandom graphs:the minimum degree condition of the H-factor when the independence number is sublinear;the minimum degree condition of the r-power of a Hamilton cycle when the independence number is sublinear;Hamilton connected and Pancyclic.Let H be an h-vertex graph and G be an n-vertex graph.An H-tiling in G is a collection of vertex-disjoint subgraphs of G isomorphic to H.An H-factor is an H-tiling which covers all vertices of G.In 2016,Balogh,Molla and Sharifzadeh determined the minimum degree condition of K3-factor when α(G)=o(n).For any r ∈ N and r≥4,Knierm and Su determined the minimum degree condition of Kr-factor when α(G)=o(n).The minimum degree conditions in the upper two results are asymptotically tight.In chapter two,for any given graph H,we give an upper bound of the minimum degree condition of H-factor.The result can be viewed an analogue of Alon-Yuster Theorem in Ramsey-Turán theory,which generalises the results of Balogh-Molla-Sharifzadeh and Knierm-Su on clique factors.In particular the degree conditions are asymptotically sharp for infinitely many graphs H which are not cliques.In 2021,Staden and Treglown conjectured that δ(G)≥(1/2+o(1))n and α(G)=o(n)are good enough to guarantee the existence of a 2-th power of a Hamilton cycle.In chapter three,we completely resolve the conjecture.Moreover we prove a general result for every r>2.The result can be viewed an analogue of Pósa-Seymour Conjecture in Ramsey-Turan theory.In particular,the minimum degree condition is also asymptotically sharp for the case r=3.In 2017,McDiarmid and Yolov proved that δ(G)≥α(G)are good enough to guarantee the existence of a Hamilton cycle.Motivated by Bondy’s Meta conjecture,in chapter four,we provide an upper bound of the minimum degree condition which is good enough to guarantee that G is Hamilton-connected and vertex-ancyclic.Recently,Draganic,Correia and Sudakov proved that δ(G)≥α(G)are good enough to guarantee that G is pancyclic,unless G is a complete bipartite graph Kn/2,n/2.