By Ian Anderson
Discrete arithmetic has now demonstrated its position in so much undergraduate arithmetic classes. This textbook presents a concise, readable and obtainable creation to a few issues during this sector, similar to enumeration, graph conception, Latin squares and designs. it truly is aimed toward second-year undergraduate arithmetic scholars, and gives them with some of the simple concepts, rules and effects. It includes many labored examples, and every bankruptcy ends with loads of routines, with tricks or ideas supplied for many of them. in addition to together with ordinary subject matters comparable to binomial coefficients, recurrence, the inclusion-exclusion precept, bushes, Hamiltonian and Eulerian graphs, Latin squares and finite projective planes, the textual content additionally contains fabric at the ménage challenge, magic squares, Catalan and Stirling numbers, and event schedules.
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Additional info for A First Course in Discrete Mathematics
Kuratowski's theorem Wh at makes a graph non-plan ar ? Clea rly, if it contains K« or K 3 •3 as a subgraph , t hen it can not possibly be planar. Itwas prov ed in 1930 by t he P olish math em atician Kurat owski th at, essent ially, it is only t he presence of a K« or a K 3 •3 within a graph that ca n stop it being plan ar. To clarify this st ate ment , we first make t he following observation. 13 cannot be plan ar eit her. For if it were, we could make a plane dr awing of it , erase b from th e edge ac, a nd obtain a plan e drawing of K «.
Lab el the ver ti ces 1, . , n + 2, an d consider t he triangle cont aining edge 12. If it cont ains verte x r as it s third vertex, in how many ways can th e remaining two pa rts of the inte rior of th e (n + 2)-gon be triangulate d? Deduc e that t n = L t itj where summation is over all pairs i , j with i+ j = n - 1. 17 Show that if 2n points are marked on the circumferen ce of a circle and if an is th e number of ways of joining t hem in pairs by n non-intersecting chords, t hen an = Cn. 4 '~;;J~ ~-2) for t he Catalan numbers , and not e that (n + I )C n = (4n - 2)Cn _ l .
Iii) ~ (i) Supp ose T is connecte d with p - 1 edges, but is not a t ree. T hen T must have a cycle . Removing an edge from a cycle does not dest roy connectedness , 50 we ca n remove edges from cycles until no cycles are left , preserving connecte dness . The resulting gra ph must be a tr ee, with p vert ices and q < p - 1 edges, cont ra dicting (ii). This th eorem can be used to est ablish t he t ree-like nature of certai n chemical molecules. 5 Show t hat t he alkanes (pa raffins) C n H2 n + 2 have tree-like molecules.
A First Course in Discrete Mathematics by Ian Anderson