Vertex Labelings and Colorings of Graphs

1. Vertex Labelings and Colorings of Graphs ValiaMitsou Graduate Center & City College City University of New York

2. introduction

3. What is a Labeling/Coloring of a Graph It is a function that maps vertices (vertex labeling), edges (edge labeling) or both vertices and edges (total labeling) of a graph G(V,E) to a set of (integer or real) numbers (or probably some other set) such that some specific property is satisfied.

4. Doesn’t seem very restrictive… Graph labelings/colorings can be used to: • express weight functions • determine a plethora of graph properties

5. Labelings vs Colorings • Labelings refer to an assignment of numbers to pieces of a graph such that a numerical property holds. • Other names: Numberings, Valuations. • Colorings most of the times don’t require an assignment of numbers (assigning colors would suffice).

6. Today… • Harmonious and Complete Colorings • Graceful Labelings

7. harmonious and complete colorings - introduction

8. Definitions 1 Franc, Harary and Plantholt (1982): • A harmonious coloring is a proper coloring of the vertices such that different color classes are connected with at most one edge. • The harmonious chromatic number h(G) of a graph G is the minimum number k for which there exists a harmonious coloring with k colors.

9. Definitions 2 Harary, Hedetniemy and Prins (1970): • A complete coloring is a proper coloring such that different color classes are connected with at least one edge. • The achromatic numberψ(G) of a graph G is the maximum number k for which there exists a complete coloring with k colors.

10. Examples Harmonious: Have enough colors such that no pair is repeated. Complete: Don’t put too many colors such that an edge cannot appear. Harmonious Coloring Complete Coloring (1,2) (1,2) 1 2 1 2 (1,3) (1,3) (1,2) (1,4) (2,3) (2,3) 2 3 4 3 (3,4) (2,3)

11. Achromatic vs Harmonious Chromatic Number G(V,E) is a graph with |E| = m. • A harmonious coloring with k colors implies that m ≤ . • A complete coloring with k colors implies that m ≥ . • ψ(G) ≤ h(G). • If ψ(G) = h(G) then m = and the coloring is called exact.

12. Achromatic vs Chromatic number χ(G) ≤ ψ(G) for any graph G. Actually we can prove that the minimum number of colors needed in a complete coloring = χ(G). • The minimum complete coloring is proper. • Suppose that an optimal proper coloring is not complete => there are two independent color classes k, l => Set k=l and obtain a proper coloring with less colors than χ(G) colors. CONTRADICTION!!!

13. Applications

14. Harmonious Colorings 1. Networking: • Machines : Vertices • Links : Edges • Assign unique ids (vertex labels) to every machine such that each pair of ids determines a unique communication link.

15. Complete Colorings Network Design – Clustering (Halldorsson): • Desired properties • Small diameter inside clusters • High connectivity outside the clusters. • Obtain a maximum complete coloring. • The classes are going to be the clusters. • Large number of classes => small size of clusters. • The partition is complete => the color classes can directly communicate with each other.

16. complexity

17. NP-Hardness Results • Determining the harmonious chromatic number or the achromatic number is shown to be NP-hard (Hopcroft - Krisnamoorthy 1983 & Yannakakis - Gavril 1980). • Both problems are shown to be NP – hard in trees, permutation graphs, quasi threshold graphs and others…

18. Kubale 2004: G(V,E) is a disconnected graph and k>0 any number. Determining whether h(G)>k is NP-complete. Proof: Reduction from independent set. G (with n vertices) has an independent set of size k iff G’ has a harmonious coloring with n+1 colors. E 2 G’ G Α A F 1 3 C Β B C 5 4 F

19. Polynomially Solvable Cases • Stars, Complete Bipartite graphs • Complete Graphs • Threshold Graphs • Paths – Cycles, Unions of Paths and Cycles Only for the harmonious chromatic number • Graphs with diameter at most 2 Only for the achromatic number: • Graphs of bounded achromatic number

20. Graphs of diameter at most 2 (harmonious chromatic number) For any node: 1

21. Graphs of diameter at most 2(harmonious chromatic number) For any node: • All nodes in distance 1 should receive distinct colors 1 1

22. Graphs of diameter at most 2(harmonious chromatic number) For any node: • All nodes in distance 1 should receive distinct colors • All nodes in distance 2 should receive distinct colors 1 3 2 1 => h(G) = n

23. Graphs of bounded achromatic number (achromatic number) • Suppose that ψ(G) = O(1). • is a constant. • Considering all the different sets of k edges which could be assigned all the different color-pairs solves the problem in time O( ). • Farber et. al. have shown that it is possible to obtain a linear time algorithm for this problem.

24. Approximability Results for the Achromatic Number • Krysta & Lorys (ESA 1999, TCS 2006): They give a 2-approximation for trees. • Kortzarz et. al. (SODA 2001, Disc. Math 2001): They give a O( ) – approx. for general graphs and an O( ) – approx. for graphs of girth at least 5. • Kortzarz & Shende (ESA 2003, Disc. Math 2008): They give a O( ) – approx. for bipartite graphs. • Kortzarz & Shende (SODA 2005): Inapproximable within a factor on bipartite graphs

25. open problems

26. Open Problems • In SODA 1997, Chaudhary & Vishwanathan conjecture that the correct approximation factor for general graphs is O( ) . (?) • Is there a PTAS for the problem defined on trees? • Study the approximability of the harmonious chromatic number problem.

27. Graceful Graphs - Introduction

28. Graceful Labeling - Definition Let G(V,E) be a graph with |E|=m edges. • Assign unique labels from the set {0, … ,m} to the vertices . • Every edge receives as a label the absolute distance of the labels of its end points. • Graceful: The produced difference edge labeling assigns unique labels 1, …, m to the edges. 1 A graceful labeling 3 4 0 4 2 2 1 3

29. Some Graceful Graph Families • Pn = Path of n edges • Sn = Star with n leaves • Cn = Cycle of n vertices (n 0 or 3 (mod 4) ) • Kn = Complete graph of n vertices (n ≤ 4) • …

30. Paths are graceful 0 n n n -1 1 n -2 n-1 2 n-2 n -5 1 n -3 n -4 … n/2

31. 1 2 n 1 n 2 3 0 n-1 3 n-1 n-2 n-2 … Stars are graceful

32. Caterpillars are graceful n-2 3 n -2 n -4 n n 0 n-3 n -3 n -7 4 … n -4 n -5 n -1 n-1 1 2

33. Cycles Cn with n = 1 or 2 (mod 4) are not Graceful • Take the sum of all the edge labels Σe = |v1 - v2| + |v2 - v3| + … |vn - v1|

34. Cycles Cn with n = 1 or 2 (mod 4) are not Graceful • Take the sum of all the edge labels Σe = |v1 - v2| + |v2 - v3| + … |vn - v1| = 2k b |a-b| |b-c| cancels In |a-b| + |b-c|, b appears as 2b appears as -2b a c

35. Cycles Cn with n = 1 or 2 (mod 4) are not Graceful • Take the sum of all the edge labels Σe = |v1 - v2| + |v2 - v3| + … |vn - v1| = 2k • The graph is Graceful => Σe = Σi = n(n+1)/2

36. Cycles Cn with n = 1 or 2 (mod 4) are not Graceful • Take the sum of all the edge labels Σe = |v1 - v2| + |v2 - v3| + … |vn - v1| = 2k • The graph is Graceful => Σe = Σi = n(n+1)/2 • n(n+1)/2 = 2k => n(n+1) = 4k => 4|n or 4|n+1 => n = 0 or 3 (mod 4)

37. RELATED PROBLEMS - APPLICATIONS

38. Ringel’s Conjecture • Definition: A decomposition of a graph G is a collection {Hi} of nonempty subgraphs such that Hi = <Ei> and {Ei} is a partition of E(G). • If Hi = H for every i, then we say that H decomposes G. H G

39. Ringel’s Conjecture • Definition: A decomposition of a graph G is a collection {Hi} of nonempty subgraphs such that Hi = <Ei> and {Ei} is a partition of E(G). • If Hi = H for every i, then we say that H decomposes G. H G

40. Ringel’s Conjecture • 1963: Ringel conjectured that every tree of order m+1 decomposes K2m+1. • 1967: Rosa proposed a hierarchy of labelings. • Among them β-valuations (which Golomb called graceful-labelings in 1972).

41. Rosa’s Hierarchy • Vertex labels among {0, …, m} • Vertex labels among {0, … 2m} • Edge labels = {1, …, m} • Edge labels = {xi : i = 1, …, m, xi = i or –i (mod 2m+1)} • There exists x (1, …, m) such that for every edge (u,v), φ(u) ≤ x and φ(v) > x. • α-valuations (= balanced labelings): 1,3,5 • β-valuations (= graceful labelings): 1,3 • σ-valuations: 2,3 • ρ-valuations: 2,4

42. Examples A ρ-valuation i: 1, 2, 3, 4, 5, 6 -i: 12, 11, 10, 9, 8, 7 An α-valuation x = 1 1 0 1 9 3 8 2 3 9 6 2 0 2 4 1 4 1 3

43. Rosa’s Theorem Theorem: A graph G with m edges decomposes K2m+1 iff it admits a ρ-valuation. Corollary: A graph admitting any valuation from the hierarchy decomposes K2m+1.

44. Rosa’s Theorem • If H is graceful having m edges then K2m+1 can be decomposed into 2m+1 copies of H. 0 6 1 0 2 3 5 2 3 2 4 3 1

45. Rosa’s Theorem • If H is graceful having m edges then K2m+1 can be decomposed into 2m+1 copies of H. 0 6 1 0 2 3 5 2 3 2 4 3 1

46. Rosa’s Theorem • If H is graceful having m edges then K2m+1 can be decomposed into 2m+1 copies of H. 0 6 1 0 2 3 5 2 3 2 4 3 1

47. Rosa’s Theorem • If H is graceful having m edges then K2m+1 can be decomposed into 2m+1 copies of H. 0 6 1 0 2 3 5 2 3 2 4 3 1

48. Rosa’s Theorem • If H is graceful having m edges then K2m+1 can be decomposed into 2m+1 copies of H. 0 6 1 0 2 3 5 2 3 2 4 3 1

49. Rosa’s Theorem • If H is graceful having m edges then K2m+1 can be decomposed into 2m+1 copies of H. 0 6 1 0 2 3 5 2 3 2 4 3 1

50. Rosa’s Theorem • If H is graceful having m edges then K2m+1 can be decomposed into 2m+1 copies of H. 0 6 1 0 2 3 5 2 3 2 4 3 1