Tiling on multipartite graphs

Transcription

1 Tiling on multipartite graphs Ryan Martin Mathematics Department Iowa State University SIAM Minisymposium on Graph Theory Joint Mathematics Meetings San Francisco, CA Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

2 1 Hajnal-Szemerédi 2 Multipartite graphs 3 Extremal examples 4 Multipartite tilings 5 Approximate bounds 6 Critical chromatic number 7 Open problems This talk includes joint work with: Csaba Magyar Endre Szemerédi, Rutgers University and the Rényi Institute Yi Zhao, Georgia State University Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

3 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

4 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. This is a K r -tiling. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

5 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. This is a K r -tiling or a K r -factor. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

6 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. This is a K r -tiling or a K r -factor or even a K r -packing. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

7 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. This is a K r -tiling or a K r -factor or even a K r -packing. We will use tiling most often. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

8 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. This is a K r -tiling or a K r -factor or even a K r -packing. We will use tiling most often. Notes r = 2 follows from Dirac Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

9 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. This is a K r -tiling or a K r -factor or even a K r -packing. We will use tiling most often. Notes r = 2 follows from Dirac r = 3 proven by Corrádi & Hajnal 1963 Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

10 The Hajnal-Szemerédi theorem Theorem (Hajnal-Szemerédi, 1970) (Complementary form) If G is a simple graph on n vertices with minimum degree ( δ(g) 1 1 ) n r then G contains a subgraph which consists of n/r vertex-disjoint copies of K r. This is a K r -tiling or a K r -factor or even a K r -packing. We will use tiling most often. Notes r = 2 follows from Dirac r = 3 proven by Corrádi & Hajnal 1963 New proof by Kierstead & Kostochka 2008 (discharging) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

11 The Alon-Yuster theorem Theorem (Alon-Yuster, 1992) For any α > 0 and graph H, there exists an n 0 = n 0 (α, H) such that in any graph G on n n 0 vertices with ( δ(g) 1 1 ) χ(h) + α n there is an H-tiling of G if V (H) divides n. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

12 The Alon-Yuster theorem Theorem (Alon-Yuster, 1992) For any α > 0 and graph H, there exists an n 0 = n 0 (α, H) such that in any graph G on n n 0 vertices with ( δ(g) 1 1 ) χ(h) + α n there is an H-tiling of G if V (H) divides n. Komlós, Sárközy and Szemerédi, 2001, showed that αn can be replaced by C = C(H), but not eliminated entirely. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

13 Multipartite graphs Definition The family of r-partite graphs with N vertices in each part is denoted G r (N). Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

14 Multipartite graphs Definition The family of r-partite graphs with N vertices in each part is denoted G r (N). Note that G G r (N) = V (G) = rn. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

15 Multipartite graphs Definition The family of r-partite graphs with N vertices in each part is denoted G r (N). Note that G G r (N) = V (G) = rn. Definition The natural bipartite subgraphs of G are the ones induced by the pairs of classes of the r-partition. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

16 Natural bipartite subgraphs Example Consider the graph G: Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

17 Natural bipartite subgraphs Example Consider the graph G: The natural bipartite subgraphs: Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

18 Minimum degree condition Definition If G G r (N), let δ(g) denote the minimum degree among all of the natural bipartite subgraphs of G. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

19 Minimum degree condition Definition If G G r (N), let δ(g) denote the minimum degree among all of the natural bipartite subgraphs of G. I.e., each vertex v V 1 has at least δ(g) neighbors in each of V 2, V 3,..., V r. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

20 Minimum degree condition Definition If G G r (N), let δ(g) denote the minimum degree among all of the natural bipartite subgraphs of G. I.e., each vertex v V 1 has at least δ(g) neighbors in each of V 2, V 3,..., V r. Conjecture [Fischer] If G G r (N) and δ(g) ( 1 1 ) N r then G has a K r -tiling. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

21 Not a corollary Observation This does not follow from Hajnal Szemerédi. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

22 Not a corollary Observation This does not follow from Hajnal Szemerédi. Let G G r (N) and δ(g) ( 1 1 r ) N. Then, Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

23 Not a corollary Observation This does not follow from Hajnal Szemerédi. Let G G r (N) and δ(g) ( 1 1 r ) N. Then, δ(g) (r 1) ( 1 1 r ) N Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

24 Not a corollary Observation This does not follow from Hajnal Szemerédi. Let G G r (N) and δ(g) ( 1 1 r ) N. Then, δ(g) (r 1) ( 1 1 r ) N = ( ) r 1 2 r (rn) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

25 Not a corollary Observation This does not follow from Hajnal Szemerédi. Let G G r (N) and δ(g) ( 1 1 r ) N. Then, δ(g) (r 1) ( 1 1 r ) N = ( ) r 1 2 r (rn) = ( 1 2r 1 ) r V (G) 2 Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

26 Not a corollary Observation This does not follow from Hajnal Szemerédi. Let G G r (N) and δ(g) ( 1 1 r ) N. Then, δ(g) (r 1) ( 1 1 r ) N = ( ) r 1 2 r (rn) = ( 1 2r 1 ) r V (G) 2 < ( 1 1 r ) V (G), if r 2. So the total degree is not large enough to invoke Hajnal-Szemerédi. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

27 Conjecture is false Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

28 Conjecture is false...but only slightly false. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

29 Conjecture is false...but only slightly false. The bound δ(g) ( 1 1 r ) N is not sufficient for (r, N) such that Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

30 Conjecture is false...but only slightly false. The bound δ(g) ( 1 1 r ) N is not sufficient for (r, N) such that r is odd, and Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

31 Conjecture is false...but only slightly false. The bound δ(g) ( 1 1 r ) N is not sufficient for (r, N) such that r is odd, and N is an odd multiple of r. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

32 Conjecture is false...but only slightly false. The bound δ(g) ( 1 1 r ) N is not sufficient for (r, N) such that r is odd, and N is an odd multiple of r. Example Let r = 3 and N = 3: Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

33 General Example Redraw the example with nonedges: Γ 3 (3) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

34 General Example Redraw the example with nonedges: Γ 3 (3) Γ r (r) This complement can be attributed to Paul Catlin, 1976, and was called a type 2 graph. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

35 Blowing up For any N, with r N, we can blow up this graph by N/r: Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

36 Blowing up For any N, with r N, we can blow up this graph by N/r: Replace each vertex with N/r vertices. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

37 Blowing up For any N, with r N, we can blow up this graph by N/r: Replace each vertex with N/r vertices. Replace each edge with K N/r,N/r. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

38 Blowing up For any N, with r N, we can blow up this graph by N/r: Replace each vertex with N/r vertices. Replace each edge with K N/r,N/r. Then, Γ r (N) G r (N). Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

39 Blowing up For any N, with r N, we can blow up this graph by N/r: Replace each vertex with N/r vertices. Replace each edge with K N/r,N/r. Then, Γ r (N) G r (N). If r N, then Γ r (N/r) has no K r -tiling iff r is odd and N/r is odd. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

40 Tripartite theorem Theorem (Magyar-M, 2002) There exists an N 0 such that if N N 0, G G 3 (N) and δ(g) 2 3 N, then G has a K 3 -tiling unless G Γ 3 (N) and N/3 is an odd integer. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

41 Tripartite theorem Theorem (Magyar-M, 2002) There exists an N 0 such that if N N 0, G G 3 (N) and δ(g) 2 3 N, then G has a K 3 -tiling unless G Γ 3 (N) and N/3 is an odd integer. (N need not be divisible by 3.) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

42 Quadripartite theorem Theorem (M-Szemerédi, 2008) There exists an N 0 such that if N N 0, G G 4 (N) and δ(g) 3 4 N, then G has a K 4 -tiling. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

43 Quadripartite theorem Theorem (M-Szemerédi, 2008) There exists an N 0 such that if N N 0, G G 4 (N) and δ(g) 3 4 N, then G has a K 4 -tiling. There is no exceptional graph. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

44 Bipartite graph tilings Theorem (Zhao, 2009) Let h be a positive integer. There exists an N 0 = N 0 (h) such that if N N 0, h N, and G G 2 (N) with then G has a K h,h -tiling. δ(g) { N 2 + h 1, N/h is odd; 2, N/h is even, N 2 + 3h 2 Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

45 Bipartite graph tilings Theorem (Zhao, 2009) Let h be a positive integer. There exists an N 0 = N 0 (h) such that if N N 0, h N, and G G 2 (N) with δ(g) = δ(g) then G has a K h,h -tiling. { N 2 N 2 + 3h 2 + h 1, N/h is odd; 2, N/h is even, Moreover, there are examples that prove that this δ condition cannot be improved. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

46 Two-colorable graph tilings Note If χ(h) = 2 and V (H) = h, then K h,h -tiling H-tiling. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

47 Two-colorable graph tilings Note If χ(h) = 2 and V (H) = h, then K h,h -tiling H-tiling. Example. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

48 Tripartite graph tilings Theorem (M-Zhao, 2009+) Let h be a positive integer and f (h) be the minimum integer such that: Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

49 Tripartite graph tilings Theorem (M-Zhao, 2009+) Let h be a positive integer and f (h) be the minimum integer such that: N 0 = N 0 (h) for which N N 0, h N, G G 3 (N), Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

50 Tripartite graph tilings Theorem (M-Zhao, 2009+) Let h be a positive integer and f (h) be the minimum integer such that: N 0 = N 0 (h) for which N N 0, h N, G G 3 (N), δ(g) h 2N 3h + f (h) implies G has a K h,h,h -tiling. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

51 Tripartite graph tilings Theorem (M-Zhao, 2009+) Let h be a positive integer and f (h) be the minimum integer such that: N 0 = N 0 (h) for which N N 0, h N, G G 3 (N), δ(g) h 2N 3h + f (h) implies G has a K h,h,h -tiling. Then f (h) = h 1, if N/h 0 mod 6; h 2 f (h) h 1, if N/h 0 mod 3; h 1 f (h) 2h 1, if N/h 3 mod 6. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

52 Tripartite graph tilings Theorem (M-Zhao, 2009+) Let h be a positive integer and f (h) be the minimum integer such that: N 0 = N 0 (h) for which N N 0, h N, G G 3 (N), δ(g) h 2N 3h + f (h) implies G has a K h,h,h -tiling. Then f (h) = h 1, if N/h 0 mod 6; h 2 f (h) h 1, if N/h 0 mod 3; h 1 f (h) 2h 1, if N/h 3 mod 6. Note Both χ(h) = 3 and V (H) = h together imply a H-tiling also. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

53 Other graph tilings The case analysis required to prove that δ(g) (3/4 + ε)n is sufficient for a K h,h,h,h -tiling would be long and difficult, using current methods. However, we believe it could be done. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

54 Other graph tilings The case analysis required to prove that δ(g) (3/4 + ε)n is sufficient for a K h,h,h,h -tiling would be long and difficult, using current methods. However, we believe it could be done. To prove the existence of an f (h) would be even more difficult. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

55 Other graph tilings The case analysis required to prove that δ(g) (3/4 + ε)n is sufficient for a K h,h,h,h -tiling would be long and difficult, using current methods. However, we believe it could be done. To prove the existence of an f (h) would be even more difficult. We cannot come close to proving that δ(g) (4/5 + ε)n is sufficient for a K 5 -tiling. Current methods break down in the quintipartite case. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

56 Best general bound Theorem (Csaba-Mydlarz, 2009+) Let r 5 and ε > 0. There exists an N 0 = N 0 (r, ε) such that if N N 0, G G r (N) and if ( ) k δ(g) k ε N, k = r + 4h r, then G has a K r -tiling. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

57 Best general bound Theorem (Csaba-Mydlarz, 2009+) Let r 5 and ε > 0. There exists an N 0 = N 0 (r, ε) such that if N N 0, G G r (N) and if ( ) k δ(g) k ε N, k = r + 4h r, then G has a K r -tiling. h r = r Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

58 Best general bound Theorem (Csaba-Mydlarz, 2009+) Let r 5 and ε > 0. There exists an N 0 = N 0 (r, ε) such that if N N 0, G G r (N) and if ( ) k δ(g) k ε N, k = r + 4h r, then G has a K r -tiling. h r = r This is the best bound for r 5. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

59 Critical chromatic number Definition Let H be a graph with order: h = V (H) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

60 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

61 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ is the size of the smallest color class of H among all proper χ-colorings of V (H). Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

62 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ is the size of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, χ cr (H) is χ cr (H) = (χ 1)h h σ. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

63 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ is the size of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, χ cr (H) is χ cr (H) = (χ 1)h h σ Fact For any graph H: χ(h) 1 < χ cr (H) χ(h) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

64 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ is the size of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, χ cr (H) is χ cr (H) = (χ 1)h h σ Fact For any graph H: χ(h) 1 < χ cr (H) χ(h) Also, χ cr (H) = χ(h) iff every proper χ-coloring of H is a equipartition. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

65 Critical chromatic number Definition Let H be a graph with order: h = V (H) chromatic number: χ = χ(h) σ is the size of the smallest color class of H among all proper χ-colorings of V (H). The critical chromatic number of H, χ cr (H) is χ cr (H) = (χ 1)h h σ Fact For any graph H: χ(h) 1 < χ cr (H) χ(h) Also, χ cr (H) = χ(h) iff every proper χ-coloring of H is a equipartition. χ cr (H) was defined by Komlós, Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

66 Use of critical chromatic number Theorem (Lower bound: Komlós, 2000) For every H and every n, divisible by V (H), there exists a G of order n with NO H-tiling and ( δ(g) = 1 1 ) n 1. χ cr (H) Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

67 Use of critical chromatic number Theorem (Lower bound: Komlós, 2000) For every H and every n, divisible by V (H), there exists a G of order n with NO H-tiling and ( δ(g) = 1 1 ) n 1. χ cr (H) Theorem (Upper bound: Komlós, 2000) For every H and ε > 0, there exists n 0 = n 0 (H, ε) such that if G has order n n 0 and ( δ(g) 1 1 ) n χ cr (H) then G has an H-tiling that covers all but εn vertices in G. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

68 Use of critical chromatic number Theorem (Lower bound: Komlós, 2000) For every H and every n, divisible by V (H), there exists a G of order n with NO H-tiling and ( δ(g) = 1 1 ) n 1. χ cr (H) Theorem (Upper bound: Komlós, 2000) For every H and ε > 0, there exists n 0 = n 0 (H, ε) such that if G has order n n 0 and ( δ(g) 1 1 ) n χ cr (H) then G has an H-tiling that covers all but εn vertices in G. Shokoufandeh & Zhao, 2003, improved εn to C = C(H). Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

69 Use of critical chromatic number Theorem (Upper bound: Komlós, 2000) For every H and ε > 0, there exists n 0 = n 0 (H, ε) such that if G has order n n 0 and ( δ(g) 1 1 ) n χ cr (H) then G has an H-tiling that covers all but εn vertices in G. Shokoufandeh & Zhao, 2003, improved εn to C = C(H). Kühn & Osthus, 2009, gave a characterization of many H for which ( δ(g) 1 1 ) n + C χ cr (H) guarantees a perfect H-tiling for C = C (H). Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

70 Open problems Given a bipartite graph H, what is the minimum degree required to ensure an H-tiling in a bipartite graph, with appropriate divisibility conditions? Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

71 Open problems Given a bipartite graph H, what is the minimum degree required to ensure an H-tiling in a bipartite graph, with appropriate divisibility conditions? I.e., (1/2 + ε)n is sufficient. What about (1 1/χ cr (H) + ε)n? Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

72 Open problems Given a bipartite graph H, what is the minimum degree required to ensure an H-tiling in a bipartite graph, with appropriate divisibility conditions? I.e., (1/2 + ε)n is sufficient. What about (1 1/χ cr (H) + ε)n? Is it true that C such that G G 5 (N) and δ(g) (4/5)N implies that there exists a partial K 5 -tiling of size (1 ε)n? Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

73 Open problems Given a bipartite graph H, what is the minimum degree required to ensure an H-tiling in a bipartite graph, with appropriate divisibility conditions? I.e., (1/2 + ε)n is sufficient. What about (1 1/χ cr (H) + ε)n? Is it true that C such that G G 5 (N) and δ(g) (4/5)N implies that there exists a partial K 5 -tiling of size (1 ε)n? Is it true that, ε > 0, N 0 = N 0 (ε) such that N N 0, G G 5 (N) and δ(g) (4/5 + ε)n implies a K 5 -tiling? Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

74 Open problems Is it true that C such that G G 5 (N) and δ(g) (4/5)N implies that there exists a partial K 5 -tiling of size (1 ε)n? Is it true that, ε > 0, N 0 = N 0 (ε) such that N N 0, G G 5 (N) and δ(g) (4/5 + ε)n implies a K 5 -tiling? Almost-covering question (r = 5) Does there exist an absolute constant C such that: For all ε > 0, if G G 5 (N), ( ) 4 δ(g) 5 + ε N and T 0 is a partial K 5 -tiling of G with T 0 < N C, then a partial K 5 -tiling T with T > T 0? Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

75 Possible solution techniques Ideas from the Kierstead-Kostochka proof Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

76 Possible solution techniques Ideas from the Kierstead-Kostochka proof e.g., discharging Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

77 Possible solution techniques Ideas from the Kierstead-Kostochka proof e.g., discharging Ideas from the Csaba-Mydlarz proof Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

78 Possible solution techniques Ideas from the Kierstead-Kostochka proof e.g., discharging Ideas from the Csaba-Mydlarz proof there is a structure that might be modified to apply their main lemma. Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23

79 Bibliography D52 G.A. Dirac, Some theorem on abstract graphs. Proc. London Math. Soc. 2 (1952), no. 3, CH63 K. Corrádi and A. Hajnal, On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hungar., 14 (1963) HSz70 A. Hajnal and E. Szemerédi, Proof of a conjecture of P. Erdős. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), North-Holland, Amsterdam, C76 P.A. Catlin, Embedding subgraphs and coloring graphs under extremal degree conditions. Ph.D. Dissertation, Ohio State University, C80 P.A. Catlin, On the Hajnal-Szemerédi theorem on disjoint cliques. Utilitas Mathematica 17 (1980), AY92 N. Alon and R. Yuster, Almost H-factors in dense graphs. Graphs Combin., 8 (1992), no. 2, F99 E. Fischer, Variants of the Hajnal-Szemerédi theorem. J. Graph Theory 31 (1999), no. 4, K00 J. Komlós, Tiling Turan theorems. Combinatorica 20 (2000), no. 2, KSSz01 J. Komlós, G.N. Sárközy and E. Szemerédi, Proof of the Alon-Yuster conjecture. Combinatorics (Prague, 1998). Discrete Math. 235 (2001), no. 1-3, MM02 Cs. Magyar and R. Martin, Tripartite version of the Corradi-Hajnal theorem. Discrete Math. 254 (2002), no. 1-3, SZ03 A. Shokoufandeh and Y. Zhao, Proof of a tiling conjecture of Komlós. Random Structures Algorithms 23 (2003), no. 2, MSz08 R. Martin and E. Szemerédi, Quadripartite version of the Hajnal-Szemerédi theorem. Discrete Math. 308 (2008), no. 19, Z09 Y. Zhao, Bipartite graph tiling. SIAM J. Discrete Math. 23 (2009), no 2, KK08 H. Kierstead and A. Kostochka, A short proof of the Hajnal-Szemerédi theorem on equitable colouring. Combin. Probab. Comput. 17 (2008), no. 2, KO09 D. Kühn and D. Osthus, The minimum degree threshold for perfect graph packings. Combinatorica, 29 (2009), no. 1, CsM09+ B. Csaba and M. Mydlarz, Approximate multipartite version of the Hajnal-Szemerédi theorem. MZ09 R. Martin and Y. Zhao, Tiling tripartite graphs with 3-colorable graphs. Electron. J. Combin, 16 (2009), no. 1, Research Paper 109, 16pp. (electronic). MZ09+ R. Martin and Y. Zhao, Tiling tripartite graphs with 3-colorable graphs: The extreme case, submitted. Survey D. Kühn and D. Osthus, Embedding large subgraphs into dense graphs. Surveys in Combinatorics 2009 (ed. S. Huczynka, J. Mitchell, C. Roney-Dougal), Cambridge University Press, 2009, Ryan Martin (Iowa State U.) Tiling on multipartite graphs 15 January / 23