Colouring powers of graphs with one cycle length forbidden

Transcription

1 Colouring powers of graphs with one cycle length forbidden Ross J. Kang Radboud University Nijmegen CWI Networks & Optimization seminar 1/2017 Joint work with François Pirot.

2 Example 1: ad hoc frequency assignment Image credit: Mesoderm/Wikipedia

3 Example 1: ad hoc frequency assignment Image credit: Mesoderm/Wikipedia

4 Example 1: ad hoc frequency assignment Image credit: Mesoderm/Wikipedia

5 Example 1: ad hoc frequency assignment How many channels needed? Image credit: Mesoderm/Wikipedia

6 Example 2: small-world networks } d neighbours

7 Example 2: small-world networks } d neighbours distance t

8 Example 2: small-world networks } d neighbours distance t How large can the network be?

9 Graph powers Given a graph G, G

10 Graph powers Given a graph G, the t-th power G t is formed from G by adding all edges between vertices at distance t. G

11 Graph powers Given a graph G, the t-th power G t is formed from G by adding all edges between vertices at distance t. G 2

12 Graph powers Given a graph G, the t-th power G t is formed from G by adding all edges between vertices at distance t. G 3

13 Graph powers Given a graph G, the t-th power G t is formed from G by adding all edges between vertices at distance t. G 4

14 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? (diameter t G t has all possible edges) Image credit: Bela Mulder/Wikipedia

15 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? (diameter t G t has all possible edges) Moore bound : G 1 + d + d(d 1) + + d(d 1) t 1 t = 1 + d (d 1) i 1 i=1 Image credit: Bela Mulder/Wikipedia

16 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? t Moore bound: G 1 + d (d 1) i 1. i=1

17 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? Moore bound: G 1 + d t (d 1) i 1. i=1 Theorem (Hoffmann & Singleton 1960) For t = 2, there are three or four graphs attaining the Moore bound. For t = 3, there can be only one. Is there a graph of diameter 2, maximum degree 57, and 3250 vertices?

18 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? t Moore bound: G 1 + d (d 1) i 1 i=1

19 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? t Moore bound: G 1 + d (d 1) i 1 d t as d. i=1

20 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? Moore bound: G 1 + d t (d 1) i 1 d t as d. i=1 Conjecture (Bollobás 1980) Fix t 1. Given ε > 0, for arbitrarily many d, there must be a graph with diameter t, maximum degree d and (1 ε)d t vertices.

21 Example 2: degree diameter problem With maximum degree d, how large can a network of diameter t be? Moore bound: G 1 + d t (d 1) i 1 d t as d. i=1 Conjecture (Bollobás 1980) Fix t 1. Given ε > 0, for arbitrarily many d, there must be a graph with diameter t, maximum degree d and (1 ε)d t vertices. Known only for t {1, 2, 3, 5}. For other choices of t, De Bruijn graphs give 0.5 t instead of 1 ε, and best known constant is t (Canale & Gómez 2005).

22 Example 1: strong edge-colouring If nearby transmissions interfere, how many channels needed?

23 Example 1: strong edge-colouring If nearby transmissions interfere, how many channels needed? Model ad hoc network with a graph. Each transmission occurs on an edge. Represent each channel by a colour. Interference at distance 2.

24 Example 1: strong edge-colouring If nearby transmissions interfere, how many channels needed? Model ad hoc network with a graph. Each transmission occurs on an edge. Represent each channel by a colour. Interference at distance 2. Problem translates: What is the least number of colours required so that edges within distance 2 must get distinct colours? Called strong chromatic index of graph.

25 Example 1: strong edge-colouring What is the least number of colours required so that edges within distance 2 must get distinct colours? The line graph L(G) of a graph G has vertices corresponding to G-edges and edges if the two corresponding G-edges have a common G-vertex.

26 Example 1: strong edge-colouring What is the least number of colours required so that edges within distance 2 must get distinct colours? The line graph L(G) of a graph G has vertices corresponding to G-edges and edges if the two corresponding G-edges have a common G-vertex.

27 Example 1: strong edge-colouring What is the least number of colours required so that edges within distance 2 must get distinct colours? The line graph L(G) of a graph G has vertices corresponding to G-edges and edges if the two corresponding G-edges have a common G-vertex.

28 Example 1: strong edge-colouring What is the least number of colours required so that edges within distance 2 must get distinct colours? The line graph L(G) of a graph G has vertices corresponding to G-edges and edges if the two corresponding G-edges have a common G-vertex. strong edge-colouring in G strong chromatic index of G

29 Example 1: strong edge-colouring What is the least number of colours required so that edges within distance 2 must get distinct colours? The line graph L(G) of a graph G has vertices corresponding to G-edges and edges if the two corresponding G-edges have a common G-vertex. strong edge-colouring in G vertex-colouring in (L(G)) 2 strong chromatic index of G chromatic number of (L(G)) 2

30 Erdős Nešetřil conjecture With maximum degree d how large can strong chromatic index be?

31 Erdős Nešetřil conjecture With maximum degree d how large can strong chromatic index be? Must be 2d(d 1) + 1 = 2d 2 2d + 1. Greedy.

32 Erdős Nešetřil conjecture With maximum degree d how large can strong chromatic index be? Must be 2d(d 1) + 1 = 2d 2 2d + 1. Greedy.

33 Erdős Nešetřil conjecture With maximum degree d how large can strong chromatic index be? Must be 2d(d 1) + 1 = 2d 2 2d + 1. Greedy. Lower bound examples? Better upper bound?

34 Erdős Nešetřil conjecture With maximum degree d how large can strong chromatic index be? d/2 Can be 5d 2 /4, d even: d/2 d/2 d/2 d/2

35 Erdős Nešetřil conjecture With maximum degree d how large can strong chromatic index be? d/2 Can be 5d 2 /4, d even: d/2 d/2 d/2 d/2 Conjecture (Erdős & Nešetřil 1980s) Must be 5d 2 /4. Theorem (Molloy & Reed 1997) Must be (2 ε)d 2 for some absolute ε > 0. (ε )

36 Colouring powers / distance colouring Let G have maximum degree d. Chromatic number of (L(G)) 2? strong chromatic index χ s(g) of G

37 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G

38 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. G if G t is a clique? degree diameter problem Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G

39 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Clique number of G t? distance-t clique number of G Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G

40 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Chromatic number of G t? distance-t chromatic number χ t(g) of G Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G

41 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Chromatic number of G t? distance-t chromatic number χ t(g) of G Greedy: must be d t + 1. Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G

42 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Chromatic number of G t? distance-t chromatic number χ t(g) of G Greedy: must be d t + 1. Conjecture: given ε > 0, can be (1 ε)d t for arbitrarily many d. Known for t {1, 2, 3, 5}. Can be t d t. Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G

43 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Chromatic number of G t? distance-t chromatic number χ t(g) of G Greedy: must be d t + 1. Conjecture: given ε > 0, can be (1 ε)d t for arbitrarily many d. Known for t {1, 2, 3, 5}. Can be t d t. Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G Greedy: must be 2d t.

44 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Chromatic number of G t? distance-t chromatic number χ t(g) of G Greedy: must be d t + 1. Conjecture: given ε > 0, can be (1 ε)d t for arbitrarily many d. Known for t {1, 2, 3, 5}. Can be t d t. Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G Greedy: must be 2d t. Conjecture: given ε > 0, must be (1 + ε)d t for all large d, if t 2. Conjecture: given ε > 0, can be (1 ε)d t for arbitrarily many d. Known for t {1, 2, 3, 4, 6}. Can be 0.5 t d t.

45 Colouring powers / distance colouring Fix t 1. Let G have maximum degree d. Chromatic number of G t? distance-t chromatic number χ t(g) of G Greedy: must be d t + 1. Conjecture: given ε > 0, can be (1 ε)d t for arbitrarily many d. Known for t {1, 2, 3, 5}. Can be t d t. Chromatic number of (L(G)) t? distance-t chromatic index χ t(g) of G Greedy: must be 2d t. Conjecture: given ε > 0, must be (1 + ε)d t for all large d, if t 2. Conjecture: given ε > 0, can be (1 ε)d t for arbitrarily many d. Known for t {1, 2, 3, 4, 6}. Can be 0.5 t d t. Kaiser & K (2014): must be (2 ε)d t for some absolute ε > 0.

46 The main questions Fix t 1. Let (G) denote the maximum degree in a graph G. What is the worst value among those G with (G) = d?

47 The main questions Fix t 1. Let (G) denote the maximum degree in a graph G. What is the worst value among those G with (G) = d? That is, What is χ t(d) := sup{χ t(g) = χ(g t ) (G) = d}? What is χ t(d) := sup{χ t(g) = χ(l(g) t ) (G) = d}?

48 The main questions Fix t 1. Let (G) denote the maximum degree in a graph G. What is the worst value among those G with (G) = d? That is, What is χ t(d) := sup{χ t(g) = χ(g t ) (G) = d}? Θ(d t ). What is χ t(d) := sup{χ t(g) = χ(l(g) t ) (G) = d}? Θ(d t ).

49 The main questions Fix t 1. Let C l denote a cycle of length l. Let (G) denote the maximum degree in a graph G. That is, What is the worst value among those G with (G) = d and no cycle C l as a subgraph? What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }? What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }?

50 The main questions Fix t 1. Let C l denote a cycle of length l. Let (G) denote the maximum degree in a graph G. That is, What is the worst value among those G with (G) = d and no cycle C l as a subgraph? What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }? What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }? We re satisfied with being correct only up to a constant factor.

51 Chromatic number, triangle-free (χ 1, l = 3) What is χ 1,3(d) := sup{χ 1(G) = χ(g) (G) = d, G C 3}? This was a question of Vizing from the 1960s (maybe motivated by Grötzsch s),

52 Chromatic number, triangle-free (χ 1, l = 3) What is χ 1,3(d) := sup{χ 1(G) = χ(g) (G) = d, G C 3}? This was a question of Vizing from the 1960s (maybe motivated by Grötzsch s), eventually settled asymptotically with the semi-random method. Theorem (Johansson 1996) χ 1,3(d) = Θ(d/ log d) as d.

53 Strong chromatic index, C 4 -free (χ 2, l = 4) What is χ 2,4(d) := sup{χ 2(G) = χ(l(g) 2 ) (G) = d, G C 4}? (Note χ 1,l (d) {d, d + 1} always.)

54 Strong chromatic index, C 4 -free (χ 2, l = 4) What is χ 2,4(d) := sup{χ 2(G) = χ(l(g) 2 ) (G) = d, G C 4}? (Note χ 1,l (d) {d, d + 1} always.) Also with the semi-random method: Theorem (Mahdian 2000) χ 2,4(d) = Θ(d 2 / log d) as d.

55 Strong chromatic index, C 4 -free (χ 2, l = 4) What is χ 2,4(d) := sup{χ 2(G) = χ(l(g) 2 ) (G) = d, G C 4}? (Note χ 1,l (d) {d, d + 1} always.) Also with the semi-random method: Theorem (Mahdian 2000) χ 2,4(d) = Θ(d 2 / log d) as d. Vu (2002) extended this to hold also for l > 4 even. The complete d-regular bipartite graph satisfies χ 2(K d,d ) = d 2, so cannot hold for any odd l.

56 Squared chromatic number, C 6 -free (χ 2, l = 6) What is χ 2,6(d) := sup{χ 2(G) = χ(g 2 ) (G) = d, G C 6}?

57 Squared chromatic number, C 6 -free (χ 2, l = 6) What is χ 2,6(d) := sup{χ 2(G) = χ(g 2 ) (G) = d, G C 6}? Theorem (K & Pirot 2016, cf. Alon & Mohar 2002) χ 2,6(d) = Θ(d 2 / log d) as d.

58 Squared chromatic number, C 6 -free (χ 2, l = 6) What is χ 2,6(d) := sup{χ 2(G) = χ(g 2 ) (G) = d, G C 6}? Theorem (K & Pirot 2016, cf. Alon & Mohar 2002) χ 2,6(d) = Θ(d 2 / log d) as d. The point-line incidence graph of a finite projective plane of order d 1 is a d-regular, girth 6 graph whose square is covered by two (d 2 d + 1)-cliques. So, if l 5, then χ 2,l (d) d 2 as d.

59 Distance vertex-colouring with one forbidden cycle length What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }?

60 Distance vertex-colouring with one forbidden cycle length What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }? Theorem (K & Pirot 2017+) Fix positive integers t and l 3. The following hold as d. For l 2t + 2 even, χ t,l (d) = Θ(d t / log d). For t odd and l 3t odd, χ t,l (d) = Θ(d t / log d). For t even and l odd, χ t,l (d) = Θ(d t ).

61 Distance vertex-colouring with one forbidden cycle length What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }? Theorem (K & Pirot 2017+) Fix positive integers t and l 3. The following hold as d. For l 2t + 2 even, χ t,l (d) = Θ(d t / log d). For t odd and l 3t odd, χ t,l (d) = Θ(d t / log d). For t even and l odd, χ t,l (d) = Θ(d t ). Last part follows from a circular unfolding of the De Bruijn graph.

62 Distance edge-colouring with one forbidden cycle length What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }?

63 Distance edge-colouring with one forbidden cycle length What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }? Theorem (K & Pirot 2017+) Fix positive integers t 2 and l 3. The following hold as d. For l 2t even, χ t,l(d) = Θ(d t / log d). For l odd, χ t,l(d) = Θ(d t ). (Note χ 1,l (d) {d, d + 1} always.)

64 Distance edge-colouring with one forbidden cycle length What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }? Theorem (K & Pirot 2017+) Fix positive integers t 2 and l 3. The following hold as d. For l 2t even, χ t,l(d) = Θ(d t / log d). For l odd, χ t,l(d) = Θ(d t ). (Note χ 1,l (d) {d, d + 1} always.) Second part uses a special bipartite graph product operation.

65 Distance colouring with one forbidden cycle length What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }? What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }? Theorem (K & Pirot 2017+) Fix positive integers t and l 3. The following hold as d. For l 2t + 2 even, χ t,l (d) = Θ(d t / log d). For t odd and l 3t odd, χ t,l (d) = Θ(d t / log d). For t even and l odd, χ t,l (d) = Θ(d t ). If t 2, then the following hold as d. For l 2t even, χ t,l(d) = Θ(d t / log d). For l odd, χ t,l(d) = Θ(d t ).

66 Main tool A sparse colouring lemma : Theorem (Alon, Krivelevich & Sudakov 1999) There exists c > 0 such that, if Ĝ is a graph of maximum degree ˆd for which at most ( ˆd 2) /f edges span each neighbourhood, then χ(ĝ) c ˆd/ log f.

67 Main tool A sparse colouring lemma : Theorem (Alon, Krivelevich & Sudakov 1999) There exists c > 0 such that, if Ĝ is a graph of maximum degree ˆd for which at most ( ˆd 2) /f edges span each neighbourhood, then χ(ĝ) c ˆd/ log f. Apply it with Ĝ = G t or L(G) t, ˆd = 2d t and f = Ω(d ε ) for some fixed ε > 0. So it suffices to show O(d 2t ε ) edges span any neighbourhood in G t or L(G) t (under the assumed cycle length restriction for G).

68 Vertex-colouring with one forbidden cycle length What is χ 1,l (d) := sup{χ 1(G) = χ(g) (G) = d, G C l }? Proposition Fix l 3. Then χ 1,l (d) = Θ(d/ log d) as d.

69 Vertex-colouring with one forbidden cycle length What is χ 1,l (d) := sup{χ 1(G) = χ(g) (G) = d, G C l }? Proposition Fix l 3. Then χ 1,l (d) = Θ(d/ log d) as d. Theorem (Erdős & Gallai 1959) Fix k. The maximum number of edges in a graph on n vertices with no path P k of length k as a subgraph is at most (k 1)n/2.

70 Vertex-colouring with one forbidden cycle length What is χ 1,l (d) := sup{χ 1(G) = χ(g) (G) = d, G C l }? Proposition Fix l 3. Then χ 1,l (d) = Θ(d/ log d) as d. Theorem (Erdős & Gallai 1959) Fix k. The maximum number of edges in a graph on n vertices with no path P k of length k as a subgraph is at most (k 1)n/2. Proof of Proposition. WLOG assume l > 3. Take any G with (G) = d, G C l and let x G.

71 Vertex-colouring with one forbidden cycle length What is χ 1,l (d) := sup{χ 1(G) = χ(g) (G) = d, G C l }? Proposition Fix l 3. Then χ 1,l (d) = Θ(d/ log d) as d. Theorem (Erdős & Gallai 1959) Fix k. The maximum number of edges in a graph on n vertices with no path P k of length k as a subgraph is at most (k 1)n/2. Proof of Proposition. WLOG assume l > 3. Take any G with (G) = d, G C l and let x G. Then G[N(x)]] d, G[N(x)] P l 2, so E(G[N(x)]) (l 3)d/2 by EG.

72 Vertex-colouring with one forbidden cycle length What is χ 1,l (d) := sup{χ 1(G) = χ(g) (G) = d, G C l }? Proposition Fix l 3. Then χ 1,l (d) = Θ(d/ log d) as d. Theorem (Erdős & Gallai 1959) Fix k. The maximum number of edges in a graph on n vertices with no path P k of length k as a subgraph is at most (k 1)n/2. Proof of Proposition. WLOG assume l > 3. Take any G with (G) = d, G C l and let x G. Then G[N(x)]] d, G[N(x)] P l 2, so E(G[N(x)]) (l 3)d/2 by EG. Apply AKS with Ĝ = G, ˆd = d, f = 2d/(l 3).

73 Proving sparsity without even cycles When excluding even C l, another classic Turán-type result is useful. Theorem (Bondy & Simonovits 1974) Fix l even. The maximum number of edges in a graph on n vertices with no cycle C l as a subgraph is O(n 1+2/l ) as n. We in fact also prove and apply a special version of it suited to our needs.

74 Circular unfolding of De Bruijn graph One of the graph constructions from K & Pirot 2016: 1. Consider [d/2] t, the words of length t on alphabet [d/2]. 2. The vertex set is t copies U 0,..., U t 1 placed around a circle. 3. Join u0u i 1 i... ut 1 i i+1 mod t i+1 mod t i+1 mod t and u0 u1... ut 1 by an edge if i+1 mod t latter is one left cyclic shift of former, i.e. uj = uj+1 i j [t 2]. A full turn of the circle shifts through all t coordinates, ensuring each U i induces a clique in t th power. Each U i has 0.5 t d t vertices. The graph is d-regular and has no odd cycles if t is even.

75 Conclusion and open problems What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }? What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }? For each t, we settled these up to a constant factor except for finitely many l.

76 Conclusion and open problems What is χ t,l (d) := sup{χ t(g) = χ(g t ) (G) = d, G C l }? What is χ t,l(d) := sup{χ t(g) = χ(l(g) t ) (G) = d, G C l }? For each t, we settled these up to a constant factor except for finitely many l. Despite no manifest monotonicity, the following are natural open questions. 1. For t 1, is there a critical l e t so that χ t,l (d) = Θ(d t ) if l < l e t even, while χ t,l (d) = Θ(d t / log d) if l l e t even? 2. For t 1 odd, is there a critical l o t so that χ t,l (d) = Θ(d t ) if l < l o t odd, while χ t,l (d) = Θ(d t / log d) if l l o t odd? 3. For t 2, is there a critical l t so that χ t,l(d) = Θ(d t ) if l < l t even, while χ t,l(d) = Θ(d t / log d) if l l t even? We proved these hypothetical critical values are at most linear in t.

77 Thank you!