The Linear 5-Arboricity of Complete Multipartite Graphs

1. The Linear 5-Arboricity of Complete Multipartite Graphs Student: 林和傑 同學(Ho-Chieh Lin) Advisor: 嚴志弘 博士(Chih-Hung Yen)

2. Contents • Introduction • Known Results • Our Findings • Conclusion • Future Works

3. Introduction All graphs considered here are finite, simple, and undirected.

4. Definition • A decomposition of a graph G is a list of subgraphs of G such that each edge of graph G appears in exactly one subgraph in the list. • Example: 1 G1 G2 K4 2 H1 H2 H3

5. G1 G2 G3 Definition • A linear k-forest is a graph whose components are paths of length at most k. • The linear k-arboricity of a graph G, denoted lak(G) , is the least number of linear k-forests needed to decompose G. • Example: la5(K6)=? la5(K6)=3 K6

6. Definition In terms of the “linear k-arboricity problem,” much attention has been focused on its two extremities: • First, when k is infinite, la∞(G) thatrepresents the case when paths of each component have unlimited lengths is the ordinary linear arboricity of G, i.e. la(G). • Second, when k is 1, la1(G) is theedge chromatic number, orchromatic indexof G, i.e. χ'(G). Then certain problem is equivalent to the “edge coloring problem.”

7. Definition • A complete multipartite graphG is an m-partite graph and m ≥ 2 such that the edge uv in E(G) if and only if u and v are in different partite sets. • We write Kn1, n2, …, nm, for the complete multipartite graph with partite sets of sizes: n1, n2, …, nm. • A balanced complete multipartite graph is a complete multipartite graph with partite sets of the same size. • We write Km(n),for the balanced complete multipartite graph with m partite sets of the same sizen.

8. Known Results on lak(G)

9. Known Results –Conjecture • Conjecture. (1982, Habib & Peroche) If G is a graph with maximum degree ∆(G) and k ≥ 2, then

10. Known Results - Cubic graph

11. Known Results –Other Regular graph

12. Known Results - Tree

13. Known Results –Complete graph

14. Known Results –Complete graph

15. Known Results –Complete Bipartite Graph

16. Known Results –Complete Bipartite Graph

17. Known Results –Km(n)

18. Known Results –Planar Graph

19. Known Results

20. Our Findings on la5(Kn,n)

21. Our Findings on la5(Kn,n)-Steps • Step 1: Calculate lower bounds of la5(Kn,n). • Step 2: Propose decomposition methods to obtain upper bounds of la5(Kn,n). • Step 3: Determine the exact values of la5(Kn,n).

22. Properties of Linear k-Arboricity • Lemma. lak (G) ≥ max { }.

23. Lower Bounds of la5(Kn,n)

24. x0 x1 y0 y1 y2 Base Concept • Let G(X,Y) be a bipartite graph with partite sets X={x0,x1,…, xr-1} and Y={y0,y1,…, ys-1}. Suppose that |Y| = s ≥ r = |X|. We define the bipartite difference (B.D.) of an edge xpyqin G(X,Y) as the valueq - p (mod s). • Ex: K2,3

25. x0 x1 x2 y0 y1 y2 Base Concept • In Kn,n, a set consisting of those edges with the same bipartite difference is a perfect matching. • Note that edges of Kn,n have n distinct bipartite differences. • Example: K3,3

26. Our Findings on la5(Kn,n) • Lemma 1. If n ≥ 6 is a multiple of 3 and α in { 0, 1,…, n-5 }, then the edges of bipartite difference α, α+1, α+2,α+3, α+4 in Kn,n can form three pairwise edge-disjoint linear 5-forests. • Example. In K15,15, all edges of bipartite difference 0, 1, 2, 3, 4 can form three pairwise edge-disjoint linear 5-forests.

27. x x x x x x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 y y y y y y y y y y y y y y y 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x x x x x x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 y y y y y y y y y y y y y y y 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x x x x x x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 y y y y y y y y y y y y y y y 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 1. B.D.= 0, 1 (Red dashed lines are unused edges of B.D. 0) Example: In K15,15, three full linear 5-forests formed by edges of Bipartite Difference (B.D.) 0, 1, 2, 3, 4 Figure 2. B.D.= 2, 3 (Blue dashed lines are unused edges of B.D. 3) Figure 3. B.D.= 4 with previously unused edges

28. Lower and Upper Bounds of la5(Kn,n)

29. Form 3 linear 5-forests Form 3 linear 5-forests Form 3 linear 5-forests Our Method for n ≡ 0 (mod 15) :Take K15,15 for example. • We group edges of Bipartite Difference (B. D.) 0, 1, 2, 3, 4 as the 1st group; 5, 6, 7, 8, 9 as the 2nd group; 10, 11, 12, 13, 14 as the 3rd group. Thus 9 linear 5-forests are needed totally to decompose K15,15

30. Lower and Upper Bounds of la5(Kn,n)

31. Properties of Linear k-Arboricity • Lemma. If H is a subgraph of G, then lak(H) ≤ lak(G).

32. Lower and Upper Bounds of la5(Kn,n)

33. Lower and Upper Bounds of la5(Kn,n)

34. Form 3 linear 5-forests + 1 green edge Form 3 linear 5-forests + 1 green edge Form 1 linear 5-forests + pink edges Form 1 linear 5-forest Form 3 linear 5-forests + 1 green Form 3 linear 5-forests + 1 green Our Strategy for n ≡ 8 (mod 15) : Take K23,23 for example. • We consider edges of Bipartite Difference (B. D.) 0, 1, 2, 3, 4 as the 1st group; 5, 6, 7, 8, 9 as the 2nd group; 10, 11, 12, 13, 14 as the 3rd group; 15, 16, 17, 18, 19 as the 4th group; 20, 21 as the 2nd tolast group; and 22 as the last group.

35. The Last Linear 5-Forest of K23,23 x x x x x x x x x x x x x x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 y y y y y y y y y y y y y y y y y y y y y y y 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 In K23,23 , an unfull linear 5-forest formed by edges of B. D. 22 and all previously unused green edges and previously unused pinkedges

36. Form 3 linear 5-forests Form 3 linear 5-forests Form 3 linear 5-forests Form 1 linear 5-forest Form 1 linear 5-forest Form 3 linear 5-forests Our Strategy for n ≡ 8 (mod 15) : Take K23,23 for example. • We group edges of Bipartite Difference (B. D.) 0, 1, 2, 3, 4 as the 1st group; 5, 6, 7, 8, 9 as the 2nd group; 10, 11, 12, 13, 14 as the 3rd group; 15, 16, 17, 18, 19 as the 4th group; 20, 21 as the 2nd tolast group; and 22 as the last group. Thus 14 linear 5-forests are needed totally to decompose K23,23

37. Lower and Upper Bounds of la5(Kn,n)

38. Lower and Upper Bounds of la5(Kn,n)

39. Lower and Upper Bounds of la5(Kn,n)

40. Exact Values of la5(Kn,n)

41. Our Findings • Theorem. la5(Kn,n) = for all n.

42. Our Findings on la5(K3(n))

43. Our Findings on la5(K3(n))-Steps • Step 1: Calculate lower bounds of la5(K3(n)). • Step 2: Propose decomposition methods to obtain upper bounds of la5(K3(n)). • Step 3: Determine the exact values of la5(K3(n)).

44. Properties of Linear k-Arboricity • Lemma. lak (G) ≥ max { }.

45. Lower Bounds of la5(K3(n))

46. x x 0 1 y y 0 1 0 1 z z Base Concept x x 0 1 K3(2) y y 0 1 z z 0 1 x x 0 1

47. Our Findings on la5(K3(n)) • Lemma 1. If n ≥ 6 is even andα in { 0, 2, 4,…, n-6}, then all edges of bipartite difference α, α+1, α+2,α+3, α+4 in K3(n) can form six pairwise edge-disjoint linear 5-forests. • Example. In K3(10), all edges of bipartite difference 0, 1, 2, 3, 4 can form six pairwise edge-disjoint linear 5-forests.

48. X, Y – edges of B.D. 0, 1 Y, Z – edges ofB.D. 2, 4 Z, X – edges ofB.D. 3 Type I:The 5 B.D. are starting from an even number.Take edges of B.D. 0, 1, 2, 3, 4 in K3(10) for example. x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 X 01 Y y y y y y y y y y y 0 1 2 3 4 5 6 7 8 9 24 z z z z z z z z z z Z 0 1 2 3 4 5 6 7 8 9 3 X x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 A path of length 5 in K3(10)

49. x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 X, Y – edges of B.D. 0, 1 y y y y y y y y y y 0 1 2 3 4 5 6 7 8 9 z z z z z z z z z z 0 1 2 3 4 5 6 7 8 9 Y, Z – edges ofB.D. 2, 4 x x x x x x x x x x Z, X – edges ofB.D. 3 0 1 2 3 4 5 6 7 8 9 The 1st Linear 5-Forest of K3(10) X Y Z X A linear 5-forest (full) formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)

50. X, Y – edges of B.D. 0, 1 Y, Z – edges ofB.D. 2, 4 Z, X – edges ofB.D. 3 x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 y y y y y y y y y y 0 1 2 3 4 5 6 7 8 9 z z z z z z z z z z 0 1 2 3 4 5 6 7 8 9 x x x x x x x x x x 0 1 2 3 4 5 6 7 8 9 The 2nd Linear 5-Forest of K3(10) X Y Z X A linear 5-forest (full) formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)