Graph Theory. Dr. Saad El-Zanati, Faculty Mentor Ryan Bunge Graduate Assistant Illinois State University REU. Graph Theory

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1 Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Grph Theory Dniel Gibson, Concordi University Jckelyn Ngel, Dominicn University Benjmin Stnley, New Mexico Stte University Allison Zle, Illinois Stte University Dr. Sd El-Znti, Fculty Mentor Ryn Bunge Grdute Assistnt 2012 Illinois Stte University REU

2 Friendship Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion There is group of six people nd not everyone is friends with everyone else. Friends Adm: Ben, Cindy Ben: Adm, Cindy Cindy: Adm, Ben, Dve, Edwrd, Frnk Dve: Cindy Edwrd: Cindy, Frnk Frnk, Cindy, Edwrd How would you drw something to depict these friendships?

3 Friendship Grph Theory Gibson, Ngel, Stnley, Zle This is one wy tht these reltionships cn be digrmmed. Specil Types of Bckground Motivtion Friends F A A: B, C B: A, C C: A, B, D, E, F D: C E: C, F F: C, E E B D C

4 Wht is Grph? Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Grph A grph G is n ordered pir (V (G), E(G)), where V (G) is nonempty finite set nd E(G) is set of 2-element subsets of V (G). Vertex nd Edge Sets V (G) is clled the vertex-set of G. The elements or things in V (G) re the vertices of G. V (G) (number of vertices in grph G) is clled the order of G. E(G) is clled the edge-set of G. The elements of things in E(G) re the edges of G. E(G) (number of edges in grph G) is clled the size of G.

5 An Exmple Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion An exmple of Grph of Order 5 nd Size 6: G = ({0, 1, 2, 3, 4}, {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 4}, {3, 4}})

6 An Exmple Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion An exmple of Grph of Order 5 nd Size 6: G = ({0, 1, 2, 3, 4}, {{0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 4}, {3, 4}}) This is drwing of G:

7 More Terminology Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Definitions A vertex is incident with n edge if the edge is connected to tht edge nd vice vers. A vertex is djcent with nother vertex if they re both incident with the sme edge. An edge is djcent with nother edge if both edges re incident with the sme vertex. Coloring We often tlk bout coloring grph s vertices. Coloring is ssigning color to vertex, nd we usully do this to group the vertices into subsets. To properly color grph, no two djcent vertices re ssigned the sme color.

8 Biprtite Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Definition A grph G is biprtite if V (G) cn be colored properly so tht no two connected edges re the sme color with minimum of two colors, sy red nd blck. Therefore, every edge hs one red end-vertex nd one blck end-vertex

9 n-regulr Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Regulr The degree of vertex is the number of edges tht re connected to tht vertex, nd is denoted by deg(v). A grph G is regulr if every vertex of G hs the sme degree. If deg(v) = n for every vertex v in V (G), then G is clled n-regulr. This exmple is 3-regulr becuse the degree of ech vertex is 3.

10 Complete Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Complete A grph is complete grph if every vertex is djcent to every other vertex. In other words, there is n edge between every vertex. A complete grph with n vertices (nd n edges) is denoted K n. Complete Biprtite A complete biprtite grph with m nd n vertices in ech of the vertex subsets respectively is denoted K m,n nd hs m n edges. In K m,n, ech vertex in the bottom set is djcent to every vertex in the top set, but not djcent to ny vertices within the bottom vertex set.

11 Cycles Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Definition A cycle is connected 2-regulr grph. A cycle is determined by the number of vertices. For exmple, cycle with 8 vertices is clled n 8-Cycle or C 8. Observtions All cycles of the sme size re isomorphic (Tht is, structurlly the sme). All cycles with size multiple of two re biprtite. Exmple:

12 Trees Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Definition A tree is grph tht does not contin ny cycles. Types of Trees A pth is tree whose vertices except for two hve degree two nd those two exclusions hve degree one. A cterpillr is tree which if you chop off ll of its legs is pth. A lobster is tree which if you chop off ll of it s legs is cterpillr. A str is tree with one vertex which is djcent with every other vertex Theorem All trees re biprtite. (Skien, 1990)

13 Pths Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion

14 Cterpillrs Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion

15 Lobsters Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion

16 Strs Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion

17 Grph Lbelings Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Lbeling Let G be grph with n edges. A lbeling of G is one-to-one function from V (G) to the set of nonnegtive integers. In other words, lbeling of G is ssigning number to ech vertex. Lbeling llows us to discuss edge length. This is n exmple of lbeled grph G: Here is G plced inside K 5 :

18 Length of n edge in K n Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Definition Let V (K n ) = Z n nd plce the vertices of K n round n n-gon. The lbel of n edge {i, j} is i j. The length of {i, j} is the shortest distnce from i to j round the polygon: length({i, j}) =min({ i j, n i j }). Edge {i, j, } is wrp-round edge (denoted with *) if its length is not equl to its lbel * 4 0 3*

19 Length of n edge in K n Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Note The number of edges of length i is dependent on n If n = 2t + 1, then K n consists of n edges of length i for ech i {1, 2,..., t}. In n = 2t, then K n consists of n edges of length i for ech i {1, 2,..., t 1} nd t edges of length t (these form 1-fctor in K 2t

20 Length of n edge in K n,n Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Let V (K n,n ) = (Z n {0}) (Z n {1}). Denote edge {(i, 0), (j, 1)} in K n,n by (i, j). The length of edge (i, j) is j i if j i nd n + j i, otherwise. Note tht E(K n,n ) consists of n edges of length i for 0 i n

21 Length of n edge in K n,n Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Let V (K n,n ) = (Z n {0}) (Z n {1}). Denote edge {(i, 0), (j, 1)} in K n,n by (i, j). The length of edge (i, j) is j i if j i nd n + j i, otherwise. Note tht E(K n,n ) consists of n edges of length i for 0 i n * 4*

22 Grph Decomposition nd Designs Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion G-decomposition Let G nd H be grphs (or multigrphs) with G subgrph of H. A G-decomposition of H is prtition of the edge set of H into subgrphs isomorphic to G (clled G-blocks). (H, G)-design A G-decomposition of H is lso clled n (H, G)-design. This is n exmple of G-decomposition where H is K 4 nd G is P 4 (G is shown below):

23 Cyclic Decompositions nd Clicking Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n ) = Z n, nd let G be subgrph of K n. By clicking G, we men pplying the isomorphism i i + 1 to V (G) Cyclic Decompositions A G-decomposition of K n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic K 3 decomposition of K 7 where V (K 7 ) = {0, 1,..., 6}: (0, 1, 3)

24 Cyclic Decompositions nd Clicking Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n ) = Z n, nd let G be subgrph of K n. By clicking G, we men pplying the isomorphism i i + 1 to V (G) Cyclic Decompositions A G-decomposition of K n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic K 3 decomposition of K 7 where V (K 7 ) = {0, 1,..., 6}: (0, 1, 3) (1, 2, 4)

25 Cyclic Decompositions nd Clicking Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n ) = Z n, nd let G be subgrph of K n. By clicking G, we men pplying the isomorphism i i + 1 to V (G) Cyclic Decompositions A G-decomposition of K n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic K 3 decomposition of K 7 where V (K 7 ) = {0, 1,..., 6}: (0, 1, 3) (1, 2, 4) (2, 3, 5)

26 Cyclic Decompositions nd Clicking Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n ) = Z n, nd let G be subgrph of K n. By clicking G, we men pplying the isomorphism i i + 1 to V (G) Cyclic Decompositions A G-decomposition of K n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic K 3 decomposition of K 7 where V (K 7 ) = {0, 1,..., 6}: (0, 1, 3) (1, 2, 4) (2, 3, 5) (3, 4, 6)

27 Cyclic Decompositions nd Clicking Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n ) = Z n, nd let G be subgrph of K n. By clicking G, we men pplying the isomorphism i i + 1 to V (G) Cyclic Decompositions A G-decomposition of K n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic K 3 decomposition of K 7 where V (K 7 ) = {0, 1,..., 6}: (0, 1, 3) (1, 2, 4) (2, 3, 5) (3, 4, 6) (4, 5, 0)

28 Cyclic Decompositions nd Clicking Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n ) = Z n, nd let G be subgrph of K n. By clicking G, we men pplying the isomorphism i i + 1 to V (G) Cyclic Decompositions A G-decomposition of K n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic K 3 decomposition of K 7 where V (K 7 ) = {0, 1,..., 6}: (0, 1, 3) (1, 2, 4) (2, 3, 5) (3, 4, 6) (4, 5, 0) (5, 6, 1)

29 Cyclic Decompositions nd Clicking Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n ) = Z n, nd let G be subgrph of K n. By clicking G, we men pplying the isomorphism i i + 1 to V (G) Cyclic Decompositions A G-decomposition of K n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic K 3 decomposition of K 7 where V (K 7 ) = {0, 1,..., 6}: (0, 1, 3) (1, 2, 4) (2, 3, 5) (3, 4, 6) (4, 5, 0) (5, 6, 1) (6, 0, 2)

30 Cyclic Decompositions in Complete Biprtite Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n,n ) = (Z n {0}) (Z n {1}). By clicking G, we men pplying the isomorphism (i, j) (i + 1, j) to V (G) Cyclic Decompositions A G-decomposition of K n,n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic S 5 decomposition of K 5,

31 Cyclic Decompositions in Complete Biprtite Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n,n ) = (Z n {0}) (Z n {1}). By clicking G, we men pplying the isomorphism (i, j) (i + 1, j) to V (G) Cyclic Decompositions A G-decomposition of K n,n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic S 5 decomposition of K 5,

32 Cyclic Decompositions in Complete Biprtite Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n,n ) = (Z n {0}) (Z n {1}). By clicking G, we men pplying the isomorphism (i, j) (i + 1, j) to V (G) Cyclic Decompositions A G-decomposition of K n,n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic S 5 decomposition of K 5,

33 Cyclic Decompositions in Complete Biprtite Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n,n ) = (Z n {0}) (Z n {1}). By clicking G, we men pplying the isomorphism (i, j) (i + 1, j) to V (G) Cyclic Decompositions A G-decomposition of K n,n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic S 5 decomposition of K 5,

34 Cyclic Decompositions in Complete Biprtite Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Clicking Let V (K n,n ) = (Z n {0}) (Z n {1}). By clicking G, we men pplying the isomorphism (i, j) (i + 1, j) to V (G) Cyclic Decompositions A G-decomposition of K n,n is cyclic if clicking G preserves the G-blocks of the decomposition. A cyclic S 5 decomposition of K 5,

35 Ros s Originl Lbelings for grph G with n edges Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion ρ-lbeling Vertex lbels from {0, 1,..., 2n}; one edge of ech length. There exists cyclic (K 2n+1, G)-design if nd only if G hs ρ-lbeling.

36 Ros s Originl Lbelings for grph G with n edges Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion ρ-lbeling Vertex lbels from {0, 1,..., 2n}; one edge of ech length. There exists cyclic (K 2n+1, G)-design if nd only if G hs ρ-lbeling. σ-lbeling No wrp-rounds. Also, cyclic (K 2n+2 F, G)-designs

37 Ros s Originl Lbelings for grph G with n edges Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion ρ-lbeling Vertex lbels from {0, 1,..., 2n}; one edge of ech length. There exists cyclic (K 2n+1, G)-design if nd only if G hs ρ-lbeling. σ-lbeling No wrp-rounds. Also, cyclic (K 2n+2 F, G)-designs β-lbeling.k.. grceful. Vertex lbels from {0, 1,..., n}.

38 Ros s Originl Lbelings for grph G with n edges Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion ρ-lbeling Vertex lbels from {0, 1,..., 2n}; one edge of ech length. There exists cyclic (K 2n+1, G)-design if nd only if G hs ρ-lbeling. σ-lbeling No wrp-rounds. Also, cyclic (K 2n+2 F, G)-designs β-lbeling.k.. grceful. Vertex lbels from {0, 1,..., n}. α-lbeling G must be biprtite. All vertex lbels from one prt in the vertex prtition must be less thn those of the other prt.

39 α-lbelings Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Theorems Every cycle with size tht is multiple of 4 hs n α-lbeling. Every pth, cterpillr, nd lobster hs n α-lbeling. Conjecture The union of two cycles with size tht is 2 more thn multiple of 4 hs n α-lbeling. α-lbelings, lthough very restrictive re very useful. One of their better pplictions is they llow for something clled stretching. We denote the lrgest number in the top prtition of n α-lbeling with the Greek letter λ.

40 Exmples of Alph Lbelings Grph Theory 0 Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion

41 Exmples of Alph Lbelings Grph Theory 0 Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion λ =

42 Exmples of Alph Lbelings Grph Theory 0 Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion λ = 0 λ =

43 Exmples of Alph Lbelings Grph Theory 0 Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion λ = 0 λ = λ =

44 Exmples of Alph Lbelings Grph Theory 0 Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion λ = 0 λ = λ = λ =

45 Stretching Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion You cn use n α-lbeling s templte in order to not just decompose K 2n+1, but lso K 2nx+1 where x is ny nturl number. Definition In other words, stretching n α-lbeling llows for G-Block to decompose n infinite number of grphs. We cll this process stretching, becuse you simply stretch the edge lengths by the size of the originl grph. Exmple:

46 Stretching Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion You cn use n α-lbeling s templte in order to not just decompose K 2n+1, but lso K 2nx+1 where x is ny nturl number. Definition In other words, stretching n α-lbeling llows for G-Block to decompose n infinite number of grphs. We cll this process stretching, becuse you simply stretch the edge lengths by the size of the originl grph. Exmple:

47 Questions Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion We know tht there is n α-lbeling of the union of ny two cycles with size 4r where r is nturl number. Gee, I wonder... Is there n α-lbeling of ny two cycles with size 4s + 2 where s is nturl number? Is there n α-lbeling of the union of ny number of cycles with size 4r where r is nturl number? Cn you tke the union of multiple clssifictions of grphs with α-lbelings to generte lbeling tht will decompose complete biprtite grph?

48 Answer Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion YES!

49 Theorem Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Let G i be biprtite grph with size m i, α-lbeling f i, criticl vlue λ i, nd vertex biprtition {A i, B i } for ll i such tht 1 i n. Also, let G = G 1 G 2 G n. There exists lbeling of G such tht G cycliclly decomposes K m+1,m+1 F where F is 1-fctor of K m+1,m+1.

50 Arrngement of G i Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion If n is even Let n = 2t for some positive integer, t. Without loss of generlity we cn ssume λ 1 λ t+1 λ 2 λ t+2 λ t λ 2t.

51 Arrngement of G i Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion If n is even Let n = 2t for some positive integer, t. Without loss of generlity we cn ssume λ 1 λ t+1 λ 2 λ t+2 λ t λ 2t. If n is odd Let n = 2t 1 for some positive integer, t. Without loss of generlity we cn ssume λ 1 λ t+1 λ 2 λ t+2 λ t 1 λ 2t 1 λ t.

52 Exmple when n is even Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion For G i such tht 1 i t f i (v) = f i (v) + i 1 (λ j + 1), v B i j=1 f i (v) + i 1 (λ j + 1) + m i 1 (m j + m 2t j+1 ) m i, v A i j=1 j=

53 Exmple when n is even Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion For G i such tht t + 1 i 2t f i (v) = f i (v) + i 1 (λ j + 1), v A i j=t+1 f i (v) + i 1 (λ j + 1) + m i 1 (m j + m 2t j+1 ) m i, v B i j=t+1 j=t

54 Exmple when n is even Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion 1* * 3* 4* 5* 7* 8* 9* 10* 6* 11* 12* 14* 15* 16* 17*

55 Exmple when n is odd Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion For G i such tht 1 i t f i (v) + i 1 (λ j + 1), v B i f i (v) = j=1 f i (v) + i 1 (λ j + 1) + m i 1 (m j + m 2t j ) m i, v A i j=1 j=

56 Exmple when n is odd Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion For G i such tht t + 1 i 2t 1 f i (v) + i 1 (λ j + 1), v A i f i (v) = j=t+1 f i (v) + i 1 (λ j + 1) + m i (m j + m 2t j+1 ), v B i j=t+1 j=t

57 Exmple when n is odd Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion * 24 3* 5* 8* 29* * * 12* 28* 31* * 17* 15 4* 6* 9* 11* 18* 19* 30*

58 Corollries Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion Corollry 1 Since we re ble to do this with ny number of grphs tht dmit n α-lbeling, then we cn produce lbeling tht decomposes complete biprtite grph using the union of ny two cycles whose sizes re congruent to 0 (mod 4). Corollry 2 Our lbeling, lthough it is bsed on α-lbelings, is not n α-lbeling itself. However, it does llow for stretching. Therefore, we cn use G-Block to decompose ny K mx+1,mx+1 complete biprtite grph; where m is the size of the G-Block, nd x is ny nturl number.

59 Grph Theory Gibson, Ngel, Stnley, Zle Specil Types of Bckground Motivtion THANK YOU!