Question

(a) (1 point) Show that if every vertex of a bipartite graph with partite sets A and B has the same degree, then both of A and B have the same size (b) (1 point) State the Marriage Theorem (c) (2 points) Prove that if every vertex of a bipartite graph G has the same degree, then it contains a perfect matching, by using the Marriage Theorem

2.4 (Thank you very much :)

Answer 1

let eved vertex be o deqree k We know bipaotte graph ond the bipartiticon is is A and B, then deg) VEB VE A Aand Here we have VEA VEB degree vextex has every as B A (o sige A the B have same e. A and

Hal's Mariage Theorem Gis A an d B bi partite qrah with partit sets a Where A& Bhas the same / sige Jhat sadurates Jhen has ibtN/s)1s X

dgree k, bipastite qraphwith partite Let every vertex ot has same is a where A and B sets AI-1B Now bom Henie it subbices SA IN(S) Is A moteing we have Hall's candi tiens, veriby 1.e. Thot sadurodes A wiu lso be a an d Saturate SSA Now, Consilo S to N/s) edges brom Is These to N/s) be The number o Let Gis k-regular t edges S O incident kIN/S) Hen ce klsklNis),i.e. (N(S)1s ko wnen abitrosi So, having chosen Hae's conditon establishead have