Topics in graph theory

A graph is a system G = (V, E) consisting of a set V of vertices and a set E (disjoint from V ) of edges, together with an incidence function End : E → M2(V ), where M2(V ) is set of all 2-element sub-multisets of V . We usually write V = V (G), E = E(G), and End = EndG. For each edge e ∈ E with End(e) = {u, v}, we called u, v the end-vertices of e, and say that the edge e is incident with the vertices u, v, or the vertices u, v are incident with the edge e, or the vertices u, v are adjacent by the edge e. Sometimes it is more convenient to just write the incidence relation as e = uv. If u = v, the edge e is called a loop; if u 6= v, the edge is called a link. Two edges are said to be parallel if their end vertices are the same. Parallel edges are also referred to multiple edges. A simple graph is a graph without loops and multiple edges. When we emphasize that a graph may have loops and multiple edges, we refer the graph as a multigraph. A graph is said to be (i) finite if it has finite number of vertices and edges; (ii) null if it has no vertices, and consequently has no edges; (iii) trivial if it has only one vertex with possible loops; (iv) empty if its has no edges; and (v) nontrivial if it is not trivial. A complete graph is a simple graph that every pair of vertices are adjacent. A complete graph with n vertices is denoted by Kn. A graph G is said to be bipartite if its vertex set V (G) can be partitioned into two disjoint nonempty parts X,Y such that every edge has one end-vertex in X and the other in Y ; such a partition {X,Y } is called a bipartition of G, and such a bipartite graph is denoted by G[X,Y ]. A bipartite graph G[X,Y ] is called a complete bipartite graph if each vertex in X is joined to every vertex in Y ; we abbreviate G[X,Y ] to Km,n if |X| = m and |Y | = n. Let G be a graph. Two vertices of G are called neighbors each other if they are adjacent. For each vertex v ∈ V (G), the set of neighbors of v in G is denoted by Nv(G), the number of edges incident with v (loops counted twice) is called the degree of v in G, denoted deg (v) or deg G(v). A vertex of degree 0 is called an isolated vertex; a vertex of degree 1 is called a leaf. A graph is said to be regular if its every vertex has the same degree. A graph is said to be k-regular if its every vertex has degree k. We always have