550: MATH Graph Theory and Combinatorics
Basic concepts, denitions and notations
1
Things
Prerequisite: basic graph theory, combinatorics, algebra, and calculus. In graph theoretic notation we mostly follow the book Introduction to Graph Theory by Doug West.
2
Graphs Denition
A graph G is a pair consisting of a vertex set V (G), and an edge set E(G) V (G) . 2 If there is no confusion about the underlying graph we often just write V = V (G) and E = E(G). x and y are the endpoints of edge e = {x, y}. They are called adjacent or neighbors. e is called incident with x and y. A loop is an edge whose endpoints are equal. Multiple edges have the same set of endpoints. In the denition of a graph we dont allow loops and multiple edges. To emphasize this, we often say simple graph. When we do want to allow multiple edges or loops, we say multigraph. Remarks A multigraph might have no multiple edges or loops. Every (simple) graph is a multigraph, but not every multigraph is a (simple) graph.
3
Special graphs
Kn is the complete graph on n vertices. Kn,m is the complete bipartite graph with partite sets of sizes n and m. Pn is the path on n vertices Cn is the cycle on n vertices
4
Isomorphism of graphs
An isomorphism of G to H is a bijection f : V (G) V (H) such that uv E(G) iff f (u)f (v) E(H). If there is an isomorphism from G to H, then we say G is isomorphic to H, denoted by G = H. Claim. The isomorphism relation is an equivalence relation on the set of all graphs. An isomorphism class of graphs is an equivalence class of graphs under the isomorphism relation.
if
and only if
5
Further denitions and notation
The degree of vertex v is the number of edges incident with v. Loops are counted twice. A set of pairwise adjacent vertices in a graph is called a clique. A set of pairwise non-adjacent vertices in a graph is called an independent set. A graph G is bipartite if V (G) is the union of two (possibly empty) independent sets of G. These two sets are called the partite sets of G. The complement G of a graph G is a graph with vertex set V (G) = V (G) and edge set E(G) = V \ E(G). 2 H is a subgraph of G if V (H) V (G), E(H) E(G). We write H G. We also say G contains H and write G H. For a subset S V (G) dene G[S], the induced subgraph of G on S: V (G[S]) = S and E(G[S]) = {e E(G) : both endpoints are in S}.
6
Walks, trails, paths, and cycles
A walk is an alternating list v0, e1, v1, e2, . . . , ek , vk of vertices and edges such that for 1 i k, the edge ei has endpoints vi1 and vi. A trail is a walk with no repeated edge. A path is a walk with no repeated vertex. A u, v-walk, u, v-trail, u, v-path is a walk, trail, path, respectively, with rst vertex u and last vertex v. If u = v then the u, v-walk and u, v-trail is closed. A closed trail (without specifying the rst vertex) is a circuit. A circuit with no repeated vertex is called a cycle. The length of a walk trail, path or cycle is its number of edges.
7
Connectivity
G is connected, if there is a u, v-path for every pair u, v V (G) of vertices. Otherwise G is disconnected. Vertex u is connected vertex to v in G if there is a u, vpath. The connection relation on V (G) consists of the ordered pairs (u, v) such that u is connected to v. Claim. The connection relation is an equivalence relation. Lemma. Every u, v-walk contains a u, v-path. The connected components of G are its maximal connected subgraphs (i.e. the equivalence classes of the connection relation). An isolated vertex is a vertex of degree 0. It is a connected component on its own, called trivial connected component.
8
Cutting a graph
A cut-edge or cut-vertex of G is an edge or a vertex whose deletion increases the number of components. If M E(G), then G M denotes the graph obtained from G by the deletion of the elements of M ; V (G M ) = V (G) and E(G M ) = E(G) \ M . Similarly, for S V (G), G S obtained from G by the deletion of S and all edges incident with a vertex from S. For e E(G), G {e} is abbreviated by G e. For v E(G), G {v} is abbreviated by G v.
Proposition. An edge e is a cut-edge iff it does not belong to a cycle.
9
Leaves, trees, forests...
A graph with no cycle is acyclic. An acyclic graph is called a forest. A connected acyclic graph is a tree. A leaf (or pendant vertex) is a vertex of degree 1. A spanning subgraph of G is a subgraph with vertex set V (G). A spanning tree is a spanning subgraph which is a tree.
Examples. Paths, stars
10
Properties of trees
Lemma. T is a tree, n(T ) 2 T contains at least two leaves. Deleting a leaf from a tree produces a tree. Theorem (Characterization of trees) For an n-vertex graph G, the following are equivalent 1. G is connected and has no cycles. 2. G is connected and has n 1 edges. 3. G has n 1 edges and no cycles. 4. For each u, v V (G), G has exactly one u, vpath. Corollary. (i) Every edge of a tree is a cut-edge. (ii) Adding one edge to a tree forms exactly one cycle. (iii) Every...