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Perfect Graphs and Vertex Coloring Problem

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Perfect Graphs and Vertex Coloring Problem IAENG International Journal of Applied Mathematics, 39:2, IJAM_39_2_06 ______________________________________________________________________________________ Perfect Graphs and Vertex Coloring Problem Hac`eneAit Haddadene (1) , Hayat Issaadi (2) ¤ Abstract|The graph is perfect, if in all its induced remarked that the smallest graph with ®(G) < θ(G) is subgraphs the size of the largest clique is equal to C5 the cycle of length ¯ve. A hole is a chordless cy- the chromatic number.In 1960 Berge formulated two cle of length at least four and an anti hole is the com- conjectures about perfct graphs one stronger than plementary graph of a hole. We say that G is a Berge the other, the weak perfect conjecture was proved graph if G contains neither odd hole nor odd anti hole. in 1972 by Lovasz and the strong perfect onjecture A graph is called γ ¡ perfect(respectively ® ¡ perfet) was proved in 2003 by Chudnovsky and al.The prob- if γ(H) = !(H)(respectively ®(G) = θ(G)) for every in- lem to determine an optimal coloring of a graph is duced subgraph H of G.It was Shannon's work which NP-complete in general case.GrÄotschell and al devel- motivated Claude Berge in 1960 [7] to propose two con- oped polynomial algorithm to solve this problem for jectures: the ¯rst known as being the weak perfect graph the whole of the perfect graphs. Indeed their algo- conjecture (WCPG). rithms are not practically e±cient. Thus, the search of very e±cient polynomial algorithms to solve these 1.1 Conjecture[7] problem in the case of the perfect graphs continues A graph G is γ ¡ perfect if and only if G is ® ¡ perfet. to have a practical interest.We will review the wealth Fulkerson (1971)[9] was the ¯rst to show that WCPG of results that have appeared on these topics using was closely related to the linear programming. bichromatic, trichromatic exchange and contraction operation. Keywords: Optimal coloring, Vertex color- ing problem, Perfect graphs, Combinatorial Algorithm 1.2 Theorem (Fulkerson[9]): 1 Introduction The following statements are equivalent: All graphs considered here are ¯nite, undirected without 1. If G is γ ¡ perfect, then G is perfect. loops or multiple edges. The vertex set of a graph G is denoted by V (G) and its 2. Let A be a (0,1)-matrix such that the covering pro- edges set by E(G).A stable set is a set of vertices no two of gram yA¸w,y ¸ 0, min 1-y,has an integer solution which are adjacent. A clique is a set of vertices every pair vector y whenever w is a (0,1)-vector. Then this pro- of which are adjacent. For v 2 V , we denoted by NG(v) gram has an integer solution vector y whenever w is the neighborhood of a vertex v in a graph G. An induced a nonnegative integer vector. subgraph of G is a graph with vertex set S ⊆ V (G) and edge set comprising all the edges of G with both end Continuing the same idea, the WCPG was proved by in S,it is denoted by G[X]. The stability number ®(G) L.Lovasz (1972)[15] and is known as the perfect graph de¯ned as the largest number of pairwise nonadjacent theorem. vertices in G.The clique covering number θ(G),de¯ned as the least number of cliques which cover all the vertices of 1.3 Perfect Graph Theorem (Lovasz[15]): G. The cardinality of a largest clique in graph G is de- noted by !(G). The number γ(G) is a minimum partition Graph G is perfect if and only if G is perfect. The sec- into stable sets in graph G .The inequality ®(G) · θ(G) ond conjecture is known as the strong perfect conjecture holds trivially for all complement of a graph G: if k cliques (SCPG). cover all the vertices then no more than k vertices can be pairewise nonadjacent.Graph which satisfy this inequal- 1.4 Conjecture[7] ity with the equality sign played an important role in Claude Shannon's (1956) paper [20] concerning the zero A graph is perfect if and only if it is Berge graph. error capacity of noisy channel.In this paper, Shannon M.Chudnovsky, N.Robertson, P.Seymour and R.Thomas proved the strong perfect graph conjecture in 2003 [8] and ¤(1) LAID3 Laboratory, Faculty of Mathematics, USTHB it became the strong perfect graph theorem. University, BP 32 El-Alia, Bab-Ezzouar 16111, Algiers, Alge- The problem to determine an optimal coloring of a graph ria. email:[email protected].(2)Department of math- ematics, Science faculty, UMBB 35000, Boumerdes,Algeria. is NP-complete in general case.GrÄotschell, Lovasz and email:[email protected]. Schrijver[10 ]developed polynomial algorithm to solve (Advance online publication: 25 May 2009) IAENG International Journal of Applied Mathematics, 39:2, IJAM_39_2_06 ______________________________________________________________________________________ this problem for the whole of the perfect graphs. This to precisely those vertices of G ¡ u ¡ v which were algorithm use an alternative of the ellipsoids method for adjacent to at least one of u or v in G. We say that Guv the resolution of linear programs.the interest of the result is obtained by contracting on fu; vg. The importance of GrÄotschell and all is not algorithmic so much.Indeed of this contraction operation is that fu; vg is a 2-pair their algorithms are not practically e±cient, undoubtedly in G then the chromatic number of G is equal to the because they do not take the combinatorial structure of chromatic number of Guv, this fact yields a simple proce- perfect graphs into account. Thus, the search of very e±- dure which gives a k-coloring of Guv yields a k-coloring cient polynomial algorithms to solve these problem in the of G. As we shall see, these procedures can be used case of the perfect graphs continues to have a practical to develop fast algorithms for ¯nding an optimal coloring. interest. 2 Coloring Techniques: 3 Results: In a graph G = (V; E), a k-coloring is a mapping We try to summaries some results about coloring some C : V ! f1; 2; ¢ ¢ ¢ ; kg such that C(u) 6= C(v) for every perfect graphs by a sequential algorithms using bichro- edge uv.Note that each color class is a stable set ,hence matic exchange, trichromatic exchange and contraction a k-coloring can be thought of a partition of the vertices operation. of a graph into stable sets S1;S2; :::; Sk.The chromatic number γ(G) is the smallest k such that G admits a 3.1 Some results about coloring using con- k-coloring.A γ(G)-coloring of G is called optimal or traction vertex: minimum coloring.It exist three technique of coloring using a vertex coloring problem: We discuss here various classes of graphs G such as there The ¯rst sequential method was proposed by Ait Had- is a sequence G0 = G; G1; :::; Gj such that Gj is a clique dadene and Ma®ray in [1], let v be a vertex of a graph G for i · j ¡ 1, Gi+1 is obtained from Gi via the con- with !(G) ¸ 4 and assume G ¡ v has been !(G)-colored. traction of an even pair of Gi (G is called even contrac- Suppose that the following property holds: there exists tile). The classes discussed include triangulated graphs, a triple of distinct colors i; j; k 2 f1; 2; ¢ ¢ ¢ ;!(G)g such comparability graphs, parity graphs and clique separable that G[Si [ Sj [ Sk [ fvg] is a K4-free Berge subgraph. graphs.Proving that a class of graphs contains a graph We can apply Tucker's [21] algorithm to 3-color the which each of its induced subgraph is even contractile component of this subgraph containing v and get in this not only means that every graph in the class is perfect it way an !-coloring of G in time O(n3). We call this also suggests a natural algorithm which will ¯nd optimal operation a trichromatic exchange. We say that v is a coloring. In the next the principle sequential algorithm Tucker vertex if the previous property holds for every is the following: !-coloring of G ¡ v. The second sequential method was proposed by Meyniel in [18],For a graph G, if there is a k-coloring of G and a Algorithm Data: let G = (V; E) be a perfect graph vertex v of G-v such as either a color i misses in N(v), satisfying property P, with jV j = n or it exists a pair (i; j) of colors such as all vertices Result: the optimal coloring of G of color j in N(v) belong to a connected component of the subgraph of G induced by vertices colored by i and j, not containing any vertex of N(v) colored i ( in 1. Find an even pair (respectively a 2-pair) in G particular a vertex colored j in N(v) is not adjacent to 2. Contract all even pair in G into a clique G of size a vertex of N(v) colored i), then G is k-colorable. To i at most !(G) obtain this k-coloring , it's enough when all the colors are present in N(v), to exchange, on the connected 3. Find an optimal coloring of G . component described previously, the color j with color i ii. if G is colored i and j , i.e. Gij=G[Si [ Sj]. An (i; j) interchanges the colors i and j in the component of Gij containing x.This operation is carried out in polynomial 3.1.1 Weakly triangulated graphs: time and is called bichromatic exchange.
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