Question

Prove that the hypercube Q_n and complete bipartite graphs K_m,n (for all m ≤ n) have chromatic index n, by explicitly describing proper n-edge colorings.

Answer 1

Ans; - . We know that every vertex of No has degree'n' because of is adjescent to every vertex of Uz , similarly every Vertex of U₂ has degree 'm' because of is adja scent to every verlex of Uq. so the degree sequence of kmin Consists of ma's and nm's listed in nouo increasing order. Here, Consider if man, then degree Sequence , (my ----, m, n, ----- ;) Here, Consider if men, then degrae sequence is (no -----,-), m -----,m) in m

of cube google Qu es (Proof bspartite.) det u, be the set of those vestices of Quline, sequares of a's and I's of lengtu n). with an even number of o's similarly, let uo be the set of thote vestices at Qu with an odd number of o's. Here clearly every vertex must have either an odd number of o's and hence, u,,U2 Position (Qu) into two disjoint pasts. le 4 Possible to have an edge sy with sy, e vi ? This would of mean tham x,y differ in exactly one Position But this would emply that if one of them has an even number of ols than the other one has an odd number of os. so, there verting Commet be both from",

- This as the Contradiction, In The same way one bronce that it is not possible to have an edge with both vertices from us, so that its proved that Qu and complete bepastite graph kun are type I for all min (me=n).