Alabama Journal of Mathematics Spring/Fall 2009 Graph Coloring: History, results and open problems Vitaly I. Voloshin Troy University, Troy, AL Invited Lewis-Parker lecture at the annual meeting of AACTM; Jacksonville State University; Jacksonville, AL; February 28, 2009 Coloring theory started with the problem of coloring the lem by algebraic methods counts more than 500 research pa- countries of a map in such a way that no two countries that pers. have a common border receive the same color. If we denote The main idea of the final proof is quite simple - by in- the countries by points in the plane and connect each pair of duction on the number of vertices; but the number of cases points that correspond to countries with a common border by is huge. Though the Kempe’s proof was erroneous, his idea a curve, we obtain a planar graph. The celebrated Four Color of alternating paths and further re-coloring of the respective Problem asks if every planar graph can be colored with 4 subgraphs was used in the final proof. A path on which two colors. It seems to have been mentioned for the first time in colors alternate is called a Kempe chain. In a plane triangu- writing in an 1852 letter from A. De Morgan to W.R. Hamil- lation, a configuration is a derived subgraph with respect to ton. Nobody thought at that time that it was the beginning of a separating cycle; it is a subgraph induced by the cycle and a new theory. The first ”proof” was given by Kempe in 1879. all the vertices which are inside the cycle in the plane. It stood for more than 10 years until Heawood in 1890 found There were the following two basic steps in the proof: a mistake. Heawood proved that five colors are enough to 1. Proof that any plane triangulation contains a configura- color any map. The Four Color Problem became one of the tion from a list of unavoidable configurations. most difficult problems in Graph Theory. Besides colorings it 2. Proof that each unavoidable configuration is reducible. stimulated many other areas of graph theory. Generally, col- In 1976, Appel, Haken, and Koch, using 1200 hours of oring theory is the theory about conflicts: adjacent vertices computer time, found 1936 (!) unavoidable configurations in a graph always must have distinct colors, i.e. they are in and proved that all they are reducible. Historically, it was the a permanent conflict. If we have a ”good” coloring, then we first time when a famous mathematical problem was solved respect all the conflicts. If we have a ”bad” coloring, then we by extensive use of computers. The final accord in this have a pair of adjacent vertices colored with the same color. one-century drama occured when the result was widely an- This looks like having a geographic map where some two nounced, the paper was published in 1977, and the Univer- countries having common border are colored with the same sity of Illinois even announced it by postage meter stamp of color. Graphs are used to depict ”what is in conflict with ”four colors suffice”, a few errors were found in the original what”, and colors are used to denote the state of a vertex. So, proof(!). more precisely, coloring theory is the theory of ”partitioning Fortunately for the authors, they have been fixed. For the the sets having internal unreconcilable conflicts” because we first time, regardless the fact that there is no human being will only count ”good” colorings. who would check the entire proof (because it contains steps The most famous erroneous proof was very instructive. that most likely can never be verified by humans), the prob- On one hand, it shows an excellent example how drawings lem is considered completely solved. may be used in Graph Theory proofs. On the other hand, In 1996, Robertson, Sanders, Seymour and Thomas im- it explicitly exhibits the limits of drawings. Since the result proved the proof by finding the set of only (!) 633 reducible was considered proven until 1890, it also demonstrated the configurations. Computers, Kempe chains, and some other fact that mathematicians of those times (like nowadays) liked techniques were used in both proofs. The three consecutive writing papers MORE than reading them. But on the top of theorems - Theorem about six colors, Theorem about five all that, nobody could even imagine what a dramatic history colors and Theorem about four colors show the main feature was ahead. Formulated first in 1852 by Francis Guthrie (a of graph coloring: there is a very simple proof for six col- student (!) of de Morgan), ”proved” in 1879 by Kempe, re- ors, a relatively simple proof for five colors and an incredible futed in 1890 by Heawod, the Four Color Problem became difficult and complex proof for four colors. Thus, the Four one of the most famous problems in Discrete Mathematics in Color Problem, formulated by a student, was first ”solved” 20th century before in 1977 it became the Four Color Theo- by a lawyer, and really solved with many contributions of rem by Appel, Haken, and Koch. the very prominent mathematicians of the century. Besides many erroneous proofs (not only that by Kempe), Let us imagine year 1912. Just 63 years passed since in it generated many new directions in Graph Theory. For ex- 1849 Gauss published his second proof of the fundamental ample, only one sub-direction of chromatic polynomials in- theorem of algebra, saying that every polynomial of degree n troduced by Birkhoff in 1912 with the aim to solve the prob- with complex coefficients has precisely n roots. Though the 1 2 VITALY I. VOLOSHIN final point in the proof was not made until 1920, at that time with combinatorial methods produced many new fruitful sci- it appeared that the theory of polynomials is so powerful and entific directions of research. universal that it can solve almost any problem. Sometimes mathematics, and graph theory in particular, So, in 1912, Birkhoff, working at Princeton, published looks like a race for generalizations. In order to general- the following paper: ”A determinant formula for the num- ize graph coloring, Erdos and Hajnal in 1966 have intro- ber of ways of coloring a map”, Birkhoff (1912). In this pa- duced hypergraph colorings: the requirement ”adjacent ver- per, he introduced a function denoted by P(G; λ) which gives tices must have different colors” was generalized to ”at least the number of proper colorings of a graph G using the col- two vertices in hyperedge must have different colors”. This ors from set f1; 2; 3; ...λg. Since it is a polynomial, it was idea was extremely fruitful and led to many generalizations called the chromatic polynomial of a graph. Suppose the of graph colorings and many hypergraph classes have been chromatic number of a graph G is χ(G); it means there are discovered. The special attention was paid to bipartite hy- no colorings using 1, 2, 3, ... χ(G) − 1 colors, and there pergraphs, normal hypergraphs (related to the weak Berge is at least one coloring using χ(G) colors. It implies that perfect graph conjecture) and extension of graph coloring to P(G; 1) = P(G; 2) = ::: = P(G; χ−1) = 0 and P(G; χ) , 0. In many set systems known long ago, like block designs etc. other words, the numbers λ =1, 2, 3,... χ − 1 all are the roots However, in all such generalizations the basic combinatorial of P(G; λ) and the number λ = χ(G) is not a root. Birkhoff problem was to find the chromatic number of a respective was motivated by the colorings of maps and his basic goal hypergraph, i.e. the minimum number of colors. The prob- was the following: prove that for every map = planar graph lem of finding the maximum number of colors systematically G, P(G; 4) , 0. That meant that any counter example G to the never appeared because for any hypergraph on n vertices the four color problem must have the root λ = 4 in its chromatic coloring using n colors always existed. From this point of polynomial. It ”only” remained to investigate the behavior view, the classic coloring theory was the theory for finding of the roots (just one root!) in the chromatic polynomials of the minimum only, i.e. it was evidently asymmetric. planar graphs to solve the famous four color problem. The end to this asymmetry was put in 1993 when the con- Birkhoff and others have shown that the chromatic poly- cept of mixed hypergraph was introduced (see the search on nomial of a graph on n vertices has degree n, with leading Google ”mixed hypergraph”). Mixed hypergraph is a triple coefficient 1 and constant term 0. Furthermore, the coeffi- H = (X; C; D) with vertex set X and two families of subsets, cients alternate in sign, and the coefficient of the second term C and D, called C-edges and D-edges respectively. Proper is −m, where m is the number of edges. Generally, since coloring of H is a mapping from X into a set of λ colors in Birkhoff, the chromatic polynomial, as function in variable λ such a way that every C-edge has two vertices of the Com- was studied in the form mon color and every D-edge has two vertices of Different colors. Now the very fundamental problem of colorability Xn P(G; λ) = r (G)λ(i); (1) appeared: not every mixed hypergraph is colorable. The i structure of uncolorable mixed hypergraphs is very general.