10.2 Graph Terminology and Special Types of Graphs

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1 10.2 Grph Terminology n Speil Types of Grphs Definition 1. Two verties u n v in n unirete grph G re lle jent (or neighors) in G iff u n v re enpoints of n ege e of G. Suh n ege e is lle inient with the verties u n v n e is si to onnet u n v. To keep trk of how mny eges re inient to vertex, we mke the following efinition. Definition 2. Let G e grph n v vertex of G. The egree of v, enote eg(v), equls the numer of eges tht re inient on v, with n ege tht is loop ounte twie. The totl egree of G is the sum of the egrees of ll the verties of G. Sine n ege tht is loop is ounte twie, the egree of vertex n e otine from the rwing of grph y ounting how mny en segments of eges re inient on the vertex. A vertex of egree zero is lle isolte. It follows tht n isolte vertex is not jent to ny vertex. A vertex is pennt if n only if it hs egree one. Consequently, pennt vertex is jent to extly one other vertex. Wht o we get when we the egrees of ll the verties of grph G = (V, E)? Eh ege ontriutes two to the sum of the egrees of the verties euse n ege is inient with extly two (possily equl) verties. This mens tht the sum of the egrees of the verties is twie the numer of eges. We hve the result in Theorem 1, whih is sometimes lle the hnshking theorem (n is lso often known s the hnshking lemm), euse of the nlogy etween n ege hving two enpoints n hnshke involving two hns. Theorem 1 (THE HANDSHAKING THEOREM). If G is ny grph, then the sum of the egrees of ll the verties of G equls twie the numer of eges of G. Speifilly, if the verties of G re v 1, v 2,..., v n, where n is nonnegtive integer, then the totl egree of G = eg(v 1 ) + eg(v 2 ) + + eg(v n ) = 2 (the numer of eges of G). Proof. Let G e prtiulr ut ritrrily hosen grph, n suppose tht G hs n verties v 1, v 2,..., v n n m eges, where n is positive integer n m is nonnegtive integer. We lim tht eh ege of G ontriutes 2 to the totl egree of G. For suppose e is n ritrrily hosen ege with enpoints v i n v j. This ege ontriutes 1 to the egree of v i n 1 to the egree v j. This is true even if i = j, euse n ege tht is loop is ounte twie in omputing the egree of the vertex on whih it is inient. Therefore, e ontriutes 2 to the totl egree of G. Sine e ws ritrrily hosen, this shows tht eh ege of G ontriutes 2 to the totl egree of G. Thus Tht is, if G = (V, E) hs m eges, then the totl egree of G = 2 (the numer of eges of G). eg(v) = 2m. (Note tht this pplies even if multiple eges n loops re present.) v V Corollry. The totl egree of grph is even. Proof. By Theorem 1 the totl egree of G equls 2 times the numer of eges, whih is n integer, n so the totl egree of G is even. Proposition. In ny grph there re n even numer of verties of o egree. 1

2 Proof. Let V e n V o e the set of verties of even egree n the set of verties of o egree, respetively, in n unirete grph G = (V, E) with m eges. Then 2m = v V eg(v) = v V e eg(v) + v V o eg(v). Beuse eg(v) is even for v V e, the first term in the right-hn sie of the lst equlity is even. Furthermore, the sum of the two terms on the right-hn sie of the lst equlity is even, euse this sum is 2m. Hene, the seon term in the sum is lso even. Beuse ll the terms in this sum re o, there must e n even numer of suh terms. Thus, there re n even numer of verties of o egree. Exmple 1. Drw grph with the speifie properties or show tht no suh grph exists.. A grph with four verties of egrees 1, 1, 2, n 3. A grph with four verties of egrees 1, 1, 3, n 3. A simple grph with four verties of egrees 1, 1, 3, n 3. Solution.. No suh grph is possile. By orollry of The Hnshking Theorem, the totl egree of grph is even. But grph with four verties of egrees 1, 1, 2, n 3 woul hve totl egree of = 7, whih is o.. Let G e ny of the grphs shown elow. In eh se, no mtter how the eges re lele, eg() = 1, eg() = 1, eg() = 3, n eg() = 3.. There is no simple grph with four verties of egrees 1, 1, 3, n 3. Proof y ontrition: Suppose there were simple grph G with four verties of egrees 1, 1, 3, n 3. Cll n the verties of egree 1, n ll n the verties of egree 3. Sine eg() = 3 n G oes not hve ny loops or prllel eges (euse it is simple), there must e eges tht onnet to,, n. By the sme resoning, there must e eges onneting to,, n. But then eg() 2 n eg() 2, whih ontrits the supposition tht these verties hve egree 1. Hene the supposition is flse, n onsequently there is no simple grph with four verties of egrees 1, 1, 3, n 3. 2

3 Exmple 2. Is it possile in group of nine stuents for eh to e friens with extly five others? Solution. The nswer is no. Imgine onstruting n quintne grph in whih eh of the nine people represente y vertex n two verties re joine y n ege if, n only if, the people they represent re friens. Suppose eh of the people were friens with extly five others. Then the egree of eh of the nine verties of the grph woul e five, n so the totl egree of the grph woul e 45. But this ontrits the orollry to The Hnshking Theorem, whih sys tht the totl egree of grph is even. This ontrition shows tht the supposition is flse, n hene it is impossile for eh person in group of nine people to e friens with extly five others. A Few Speil Simple Grphs Definition 3. A yle C n, n 3, onsists of n verties v 1, v 2,..., v n n eges {v 1, v 2 }, {v 2, v 3 },..., {v n 1, v n }, n {v n, v 1 }. Exmple 3. The yles C 3, C 4, n C 5 re isplye elow Definition 4. We otin wheel W n when we n itionl vertex to yle C n, for n 3, n onnet this new vertex to eh of the n verties in C n, y new eges. Exmple 4. The wheels W 3, W 4, W 5, n W 6 re isplye elow. Exmple 5. An n-imensionl hyperue, or n-ue, enote y Q n, is grph tht hs verties representing the 2 n it strings of length n. Two verties re jent if n only if the it strings tht they represent iffer in extly one it position. We isply Q 1, Q 2, Q 3, n Q 4 elow. Note tht you n onstrut the (n + 1)-ue Q n+1 from the n-ue Q n y mking two opies of Q n, prefing the lels on the verties with 0 in one opy of Q n n with 1 in the other opy of Q n, n ing eges onneting two verties tht hve lels iffering only in the first it. In the figure ove, Q 3 is onstrute from Q 2 y rwing two opies of Q 2 s the front n k fes of Q 3, ing 0 t the 3

4 eginning of the lel of eh vertex in the k fe n 1 t the eginning of the lel of eh vertex in the front fe. (Here, y fe we men fe of ue in three-imensionl spe. Think of rwing the grph Q 3 in three-imensionl spe with opies of Q 2 s the front n k fes of ue n then rwing the projetion of the resulting epition in the plne.) Definition 5. A simple grph G is lle iprtite if its vertex set V n e prtitione into two isjoint sets V 1 n V 2 suh tht every ege in the grph onnets vertex in V 1 n vertex in V 2 (so tht no ege in G onnets either two verties in V 1 or two verties in V 2 ). When this onition hols, we ll the pir (V 1, V 2 ) iprtition of the vertex set V of G. Exmple 6. C 6 is iprtite, euse its vertex set n e prtitione into the two sets V 1 = {v 1, v 3, v 5 } n V 2 = {v 2, v 4, v 6 }, n every ege of C 6 onnets vertex in V 1 n vertex in V 2. Some grphs re lle omplete in the sense tht ll pirs of verties re onnete y eges. Definition 6. Let n e positive integer. A omplete grph on n verties, enote K n, is simple grph with n verties n extly one ege onneting eh pir of istint verties. Exmple 7. We my rw the omplete grphs K 1, K 2, K 3, K 4, K 5, n K 6 s follows: Exmple 8. K 3 is not iprtite. To verify this, note tht if we ivie the vertex set of K 3 into two isjoint sets, one of the two sets must ontin two verties. If the grph were iprtite, these two verties oul not e onnete y n ege, ut in K 3 eh vertex is onnete to every other vertex y n ege. Theorem 2. A simple grph is iprtite if n only if it is possile to ssign one of two ifferent olors to eh vertex of the grph so tht no two jent verties re ssigne the sme olor. In nother lss of grphs, the vertex set n e seprte into two susets: Eh vertex in one of the susets is onnete y extly one ege to eh vertex in the other suset, ut not to ny verties in its own suset. Suh grph is lle omplete iprtite. Definition 7. Let m n n e positive integers. A omplete iprtite grph on (m, n) verties, enote K m,n, is simple grph with istint verties v 1, v 2,..., v m n w 1, w 2,..., w n tht stisfies the following properties: For ll i, k = 1, 2,..., m n for ll j, l = 1, 2,..., n, 1. There is n ege from eh vertex v i to eh vertex w j. 2. There is no ege from ny vertex v i to ny other vertex v k. 3. There is no ege from ny vertex w j to ny other vertex w l. 4

5 Exmple 9. The omplete iprtite grphs K 3,2 n K 3,3 re illustrte elow. v 1 v 1 w 1 w 1 v 2 v 2 w 2 w 2 v 3 v 3 w 3 5