Cycles And Related Topics In Graphs

In this paper, we mainly discuss the cycles and the related topics.Firstly, we consider2-factor problems, which generalize the Hamilton cycle problems. Hamiltonian cycle plays an important role in graph theory and the2-factor problems have caused much attention. In the paper, the2-connected and K1,3-frec graphs will be discussed, the following result is proved:Let G be a2-connected K1,3-free graph of order n=|V(G)|>51and for each pair of non-adjacent vertices x and y, d(x)+d(y)≥(2n-4)/3.Then for each k with2≤k≤(n-24)/3, one of the following must hold:(1)G has a2-factor with exactly k components (2)G has a2-factor with exactly k+1components C1, C2,…Ck+1,|C1|≥|C2|≥…|Ck+1|,then|Ck+1|≥4and G[Ck+1] is completeNext, the properly colored cycle will be discussed.The coloring problem is also a major branch in Graph Theory. We mainly focus on the graph without triangles and prove the following result:Let G be an edge coloring graph without triangles such that for each ver-tex v there are at least d different colors on edge incident to v. We prove that G contains a properly colored path of length at least4d-2or a properly colored cycle of length at least2d-2.Lastly, we discuss the theory of group connectivity. In this paper, we prove the following result: Let, G be a.3-regular connected graph, then G is Z3-cormected if and only if G is graph1or graph2in the paper.