Bipartite Graphs and their ApplicationsBipartite graphs are perhaps the most basic of objects in graph theory, both from a theoretical and practical point of view. However, sometimes they have been considered only as a special class in some wider context. This book deals solely with bipartite graphs. Together with traditional material, the reader will also find many unusual results. Essentially all proofs are given in full; many of these have been streamlined specifically for this text. Numerous exercises of all standards have also been included. The theory is illustrated with many applications especially to problems in timetabling, chemistry, communication networks and computer science. For the most part the material is accessible to any reader with a graduate understanding of mathematics. However, the book contains advanced sections requiring much more specialized knowledge, which will be of interest to specialists in combinatorics and graph theory. |
Contents
| 1 | |
Introduction to bipartite graphs | 7 |
Metric properties | 23 |
Connectivity | 45 |
Maximum matchings | 56 |
Expanding properties | 75 |
Subgraphs with restricted degrees | 97 |
Edge colourings | 125 |
Common terms and phrases
a e V1 adjacency matrix adjacent algorithm Asratian augmenting path bipartite subgraph bipartition V1,V2 colour classes colour-feasible column complete bipartite graph consider construct contains Corollary da(u decomposition define degree denote doubly stochastic matrix e e E(G edge-disjoint edges incident edges joining edges of G elements exists f-factor G with bipartition graph G graph with bipartition Hamilton cycle Hamilton path induced subgraph induction isomorphic k-regular bipartite graph latin square Lemma length Let G matching of G maximum matching Menger's Theorem minimum vertex covering n-cube NP-complete number of edges obtained pair of vertices partition pebbles pendant vertices perfect matching permutation polynomial positive integer Proof Property Proposition Prove sequence Show simple bipartite graph simple graph spanning tree stable matching subgraph of G Suppose symbols T-join triangle-free graph uniquely edge colourable v e V2 vertex a e vertex set