λ -Harmonious Graph Colouring

Transcription

1 λ -Harmonious Graph Colouring Lauren DeDieu McMaster University Southwestern Ontario Graduate Mathematics Conference June 4th, 201

2 What is a graph?

3 What is vertex colouring? Figure : Proper Colouring. 2 1

4 What is vertex colouring? Figure : Proper Colouring Figure : Improper Colouring.

5 Edge Colour Pair The unordered pair of colours assigned to an edge s incident vertices e Figure : e has the colour pair {1,2}.

6 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair.

7 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G.

8 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G. Denoted by h(g).

9 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G. Denoted by h(g).

10 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G. Denoted by h(g). 1

11 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G. Denoted by h(g). 1 2

12 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G. Denoted by h(g). 1 2

13 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G. Denoted by h(g)

14 Harmonious Graph Colouring A proper colouring such that no two edges share the same colour pair. harmonious chromatic number: the least number of colours needed to harmoniously colour a graph G. Denoted by h(g) Figure : h(p 4 ) =.

15 Lower Bound for h(g) ( k ) ( 2 tells us how many ways k colours can be arranged into pairs of 2, so E(G) k ) 2.

16 Lower Bound for h(g) ( k ) ( 2 tells us how many ways k colours can be arranged into pairs of 2, so E(G) k ) 2. Let k be the least integer such that this inequality holds. Then h(g) k.

17 Lower Bound for h(g) ( k ) ( 2 tells us how many ways k colours can be arranged into pairs of 2, so E(G) k ) 2. Let k be the least integer such that this inequality holds. Then h(g) k Figure : A graph with 10 edges needs at least 5 colours.

18 Harmonious Chromatic Number h(g) has been found for several families of graphs like paths (Miller, Pritikin 91 ), cycles (Lee, Mitchem 87 ), and trees (Mitchem 89 ).

19 Harmonious Chromatic Number h(g) has been found for several families of graphs like paths (Miller, Pritikin 91 ), cycles (Lee, Mitchem 87 ), and trees (Mitchem 89 ). We will generalize h(g) to allow for up to λ edge colour pairs.

20 λ-harmonious Graph Colouring A proper colouring such that no λ + 1 edges share the same colour pair.

21 λ-harmonious Graph Colouring A proper colouring such that no λ + 1 edges share the same colour pair. λ-harmonious chromatic number: the least number of colours needed to λ-harmoniously colour a graph G.

22 λ-harmonious Graph Colouring A proper colouring such that no λ + 1 edges share the same colour pair. λ-harmonious chromatic number: the least number of colours needed to λ-harmoniously colour a graph G. Denoted by h λ (G).

23 λ-harmonious Graph Colouring A proper colouring such that no λ + 1 edges share the same colour pair. λ-harmonious chromatic number: the least number of colours needed to λ-harmoniously colour a graph G. Denoted by h λ (G). Note: We have h(g) when λ = 1.

24 λ-harmonious Graph Colouring A proper colouring such that no λ + 1 edges share the same colour pair. λ-harmonious chromatic number: the least number of colours needed to λ-harmoniously colour a graph G. Denoted by h λ (G). Note: We have h(g) when λ = Figure : h 1 (P ) =,h 2 (P ) = 2.

25 λ-harmonious Graph Colouring A proper colouring such that no λ + 1 edges share the same colour pair. λ-harmonious chromatic number: the least number of colours needed to λ-harmoniously colour a graph G. Denoted by h λ (G). Note: We have h(g) when λ = Figure : h 1 (P ) =,h 2 (P ) = 2. ( E(G) λ k ) 2

26 Complete Bipartite Graphs Complete Bipartite Graph - a graph whose vertex set can be decomposed into two disjoint sets such that no two vertices in the same set are adjacent, and every pair of vertices in distinct sets are adjacent. We will denote a complete bipartite graph by K m,n where m and n are the sizes of the disjoint sets, with m n. m n

27 Complete Bipartite Graphs The λ-harmonious chromatic number of a complete bipartite graph K m,n with m n is: Theorem : h λ (K m,n ) = min m n λq + ;1 q λ,q Z q.

28 h 4 (K 8, ): Figure : m 4 + n = 2 + = 5

29 h 4 (K 8, ): Figure : m 2 + m2 = = 6

30 h 4 (K 8, ): Figure : m 2 + m2 = = 6 h4 (K 8, ) = m 4 + n = 2 + = 5

31 h 4 (K 2,2 ): Figure : m 4 + n = =

32 h 4 (K 2,2 ): Figure : m 2 + m2 = = 2

33 h 4 (K 2,2 ): Figure : m 2 + m2 = = 2 h4 (K 2,2 ) = m 2 + m2 = = 2

34 Paths Path - an alternating sequence of distinct vertices and edges that begins and ends with a vertex. We denote a path on n vertices by P n. Figure : A path of length, denoted P 4

35 Paths The λ-harmonious chromatic number of a path, P n, is as follows: Theorem: Let r Z be determined by the inequality λ ( 2r 1) ( 2 < n 1 λ 2r+1 ) 2. Then 2r h λ (P n ) = 2r + 1 if λ is even and n 1 λ ( 2r) 2, or, if λ is odd and n 1 λ ( 2r) 2 (r 1), otherwise.

36 Proof Idea for Paths: An Eulerian path is a trail in a graph that visits each edge exactly once.

37 Proof Idea for Paths: An Eulerian path is a trail in a graph that visits each edge exactly once. A graph has an Eulerian path it is connected and has at most two vertices of odd degree.

38 Proof Idea for Paths: An Eulerian path is a trail in a graph that visits each edge exactly once. A graph has an Eulerian path it is connected and has at most two vertices of odd degree. A complete graph on n vertices, Kn, is a graph such that each pair of vertices is connected by an edge.

39 Proof Idea for Paths (λ = 1): Properly colour a complete graph K n (i.e. give each vertex a unique colour).

40 Proof Idea for Paths (λ = 1): Properly colour a complete graph K n (i.e. give each vertex a unique colour). If there exists an Eulerian path of length k in Kn, then this gives us a proper colouring of the path P k+1.

41 Proof Idea for Paths (λ = 1): Properly colour a complete graph K n (i.e. give each vertex a unique colour). If there exists an Eulerian path of length k in Kn, then this gives us a proper colouring of the path P k+1. Given a path on m vertices, P m, the harmonious chromatic number, h 1 (P m ) will be the smallest k such that there exists a complete graph K k with an Eulerian path of length m 1.

42 Example: We know ( 5 ( 2) < 11,12,1,14,15 6 ) 2 h 1 (P 12 ),h 1 (P 1 ),h 1 (P 14 ),h 1 (P 15 ),h 1 (P 16 ) 6. So, we consider K 6. Each edge has odd degree, so we must delete at least 2 edges to have an Eulerian path. After removing v and w we re left with only two vertices of odd degree, so we can trace an Eulerian path , and we can see h 1 (P 12 ) = h 1 (P 1 ) = h 1 (P 14 ) = v w Figure : K {v,w} has an eulerian path of length 1.

43 Proof Idea for Paths (λ = n): Consider a graph on k vertices in which each pair of vertices is connected by λ edges, denoted H λ,k.

44 Proof Idea for Paths (λ = n): Consider a graph on k vertices in which each pair of vertices is connected by λ edges, denoted H λ,k. Figure : H 2,5.

45 Proof Idea for Paths (λ = n): Consider a graph on k vertices in which each pair of vertices is connected by λ edges, denoted H λ,k. Figure : H 2,5. Given a path on m vertices, P m, the λ-harmonious chromatic number, h λ (P m ), will be the smallest k such that H λ,k has a subgraph with an eulerian path of length m 1.

46 Cycles Cycle - a closed path where the first and last vertex are the same. We denote a cycle on n vertices by C n. Figure : A cycle of length 6, denoted C 6

47 Cycles The λ-harmonious chromatic number of a cycle, C n, is as follows: Theorem: Let k be the least integer such that n λ ( k 2). Then h λ (C n ) = k k + 1 if one of the following four conditions hold: i) λ is even and n λ ( k 2) 1, ii) λ 1, λ is odd, k is odd, and n λ ( k 2) 1, ii) λ = 1, k is odd, and n λ ( k 2) i for i = 1,2, iv) λ is odd, k is even, and n λ ( k 2) i for i = 0... k 2 1, otherwise.,

48 Example: If we have a cycle C 7, then we know we h 1 (C 7 ) 5, since ( 4 2) < 7 < ( 5 2). So, we consider K 5. If we delete the edges from a cycle of length in K 5, then we are left with a subgraph of K 5 with an eulerian cycle of length 7. Therefore, h(c 7 ) = 5. K K 5 C Figure : An eulerian subgraph of K 5 with 7 edges.

49 Wheels Wheel - a graph on n vertices formed by connecting a single vertex to all vertices of a C n 1. We denote a wheel on n vertices as W n. Figure : A wheel on 7 vertices, denoted W 7

50 Wheels Note that each wheel contains a star K n 1,1 and a cycle C n 1. Figure : W 7

51 Wheels Note that each wheel contains a star K n 1,1 and a cycle C n 1. Figure : W 7 contains a star K 6,1

52 Wheels Note that each wheel contains a star K n 1,1 and a cycle C n 1. Figure : W 7 contains a cycle C 6

53 Wheels Theorem: Let h λ (K n 1,1 ) = n 1 λ + 1 = t. If t > 4, then h λ (W n ) = t.

54 Future Work Although h(g) has been found for several families of graphs, the harmonious colouring problem has been proven to be NP-complete (Johnson 8 ). i.e. Given a graph G and a positive integer k V(G), can G be harmoniously coloured with k colours? We suspect that the λ-harmonious colouring problem is also NP-complete, but we haven t been able to prove it yet. i.e. Given a graph G and a positive integer k V(G), can G be λ-harmoniously coloured with k colours?

55 Bibliography J.E. Hopcroft, M.S. Krishnamoorthy, On the harmonious coloring of graphs, SIAM. J. Algebraic and Discrete Methods 4 (198), S. Lee, J. Mitchem, An upper bound for the harmonious chromatic number of a graph, J. Graph Theory 11 (1987), Z. Miller, D. Pritikin, The harmonious coloring number of a graph, Discrete Mathematics 9 (1991), J. Mitchem, On the harmonious chromatic number of a graph, Discrete Mathematics 74 (1989),

56 Thank you