Lecture 1 : Matchings on bipartite graphs

An undirected graph G = (V,E) consists of a finite set V of vertices and a finite multi-set of unordered pairs E of edges. A loop is an edge of the form (v, v). When E is a proper set (not a multi-set), G is said to be simple. When E is an ordered set, the graph is said to be directed. An edge e = (u, v) ∈ E(G) is said to be incident to u and v, while u and v are adjacent. The complement of a graph G, denoted by Ḡ is the graph whose vertex set is the same as that of G, and two vertices in Ḡ are adjacent iff they are not adjacent in G. A walk is a sequence of vertices v1, . . . , vk where vivi+1 ∈ E(G). A path is a walk without repeated vertex in the sequence. A path that starts with u and end with v is called a path from u to v or a (u, v)path. The length of a path is the number of edges in the path. The distance d(u, v) between two vertices u and v is the minimum length of (u, v)-paths. Note that d(u, v) could be infinite. A cycle is a walk which starts and ends at the same vertex and all the vertices in the middle do not repeat in the walk. An n-cycle or a cycle of length n is a cycle with n edges. The girth of a graph G is the minimum length of a cycles. The degree dG(v) of a vertex v is the number of edges incident to v. A graph is regular if all vertices have the same degree. We often use ∆(G) and δ(G) to denote the maximum and minimum degree of G, respectively. A graph is k-regular if ∆(G) = δ(G) = k. A subgraph G′ = (V ′, E′) of G = (V,E) is a graph such that V ′ ⊆ V and E′ ⊆ E. An induced subgraph G′ = (V ′, E′) of G = (V,E) is a subgraph such that for any u′, v′ ∈ V ′, u′v′ ∈ E implies u′v′ ∈ E′. When V ′ = V , G′ is said to be a spanning subgraph of G. A graph G is connected if there is a path between any two vertices in G. It is disconnected otherwise. A component of a graph is a maximal connected induced subgraph. A tree is a graph with no cycle. A forest is a graph all of whose components are trees. A spanning tree of a graph G is a spanning subgraph of G which is also a tree. For any graph G and a subset V ′ of V (G), we use G− V ′ to denote the graph obtained by removing all vertices in V ′ and the edges one of whose end points in is V ′. For any subset E′ of edges, we use G− E′ to denote (V,E − E′). A subset C of vertices is called a vertex cut if G − C is not connected. A subset S of edges is said to be an edge cut if G − S is disconnected. The vertex-connectivity of G, denoted by κ(G), is the size of a minimum vertex cut. Similarly, the edge-connectivity, denoted by κ′(G), is the size of a minimum edge cut. If a vertex cut (edge cut) contains one vertex (edge) only, then the vertex (edge) is called a cut vertex (cut edge or bridge). A graph is k-connected if k ≤ κ(G), and k-edge-connected if k ≤ κ′(G). An n-factor of a graph G is an n-regular subgraph of G. A matching of G is a 2-factor of G, and it is said to be perfect if it contains all vertices of G. A bipartite graph G is a graph whose vertex set V (G) can be partitioned into two non-empty subsets X and Y . This partition is often called the bipartition of V . The sets X and Y are often called the color classes of G.