Vector spaces and the four-color problem

Brief Communication Vector Spaces and the Four-Color by

LEONARD

Problem

S. BOBROW

Department of Electrical and Computer Engineering University of Massachusetts, Amherst, Massachusetts

I. Introduction A recent generalization of the matrix representations of graphs (1)has resulted in the description of many weighted, directed graphs by matrices whose entries are elements of GF(q), the Galois field with q elements. For the special cases q = 2 and q = 3, the result is the familiar matrix descriptions of all undirected graphs and all directed graphs, respectively. In this paper, this generalization is utilized to formulate an algebraic approach to the four-color problem. Let G be a weighted, directed graph with n edges, and let the edge weights be nonzero elements in GF(q). Suppose that C is an oriented circuit of G, and suppose that the weight of edge e, is denoted by wi for i = 1,2, . . . , n. Define a “q-ary vector representation” of C by C = (cl, c2, . . . , c,J in the following manner: (1) If e,$C, then ci = 0. (2) If ei EC and the orientations of e, and C agree, then Ci = wi. (3) If e4EC and the orientations of ei and C do not agree, then ci = -wi. Next, if S is an oriented seg* of G, then define a q-ary vector representation of S by S = (s 1, s 2, . . ..s.) in the following manner: (1) If ei $ S, then Si = 0. (2) If ei ES and the orientations of ei and S agree, then si = wil. (3) If eiES and the orientations of e, and S do not agree, then si = - w;l. With these definitions, it is not difficult to prove the following lemma (1). Lemma 1. pet C be an oriented circuit and S an oriented seg of a weighted, directed graph G, and let C and S be the corresponding vector representations. Then C-S” = 0

[inner product over GF(q)],

i.e. vectors C and S are orthogonal over GF(q). The graph G may now be described by matrices whose entries are elements of GF(q). Suppose that the edges of G are renumbered such that for some tree t of G edges e,, e2, . . ., eN (N is the nullity of the graph) are in H= G - t, and eN+l, eN+2,. .., e, are in t. Define a “fundamental circuit matrix” B,(G) for G in the following manner: The ith row of B,(G) is the q-ary vector corresponding to the fundamental circuit which contains ei and whose * A seg is a cut-set or a union of edge disjoint cut-sets.

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Leonard S. Bobrow orientation agrees with the orientation of ei (; = 1,2, . . ., N). From this matrix, a “modified fundamental circuit matrix” B,(G) is obtained by multiplying the ith row of B,(G) by w;’ for i = 1,2, . . ., N. The resulting (N x n) q-ary matrix is of the form B,(G) = [IN B,,], where IN is the (N x N) identity matrix. In a similar manner, also define a “fundamental cut-set (or seg) matrix” Qf(G) and a “modified fundamental cut-set matrix” Q,(G) as follows : The ith row of Q,(G) is the q-ary vector corresponding to the fundamental cut-set which contains edge eN+i and whose orientation agrees with the orientation of eN+i (i = 1,2, . . . . R ; where R is the rank of the graph). To obtain Q,(G), multiply the ith row of Qr(G) by w~+~ for i = 1,2, . . . . R. Q,(G) and Q,,(G) are (Rx n) q-ary matrices, and Q,(G) has the form Q,(G)= [&u&l, w here I, is the (R x R) identity matrix. With the use of Lemma 1 and matrix arithmetic, it is a simple matter to show that Qrl = (- B,$. Clearly, the rows of B,(G) and the rows of Qr(G) are linearly independent and consequently form bases for linear vector spaces over the Galois field GF(q). These vector spaces, respectively designated by V, and Vi are, by Lemma 1, orthogonal. That is, V, is the null space of V, and vice versa. Next define a q-ary “incidence matrix” A as follows: If edge ei is directed from vertex 2; to vertex 9, then the ith column of A has entries wir in row j, -Wi -l in row 1 and 0 everywhere else. If the graph has n edges and m vertices, then A is an (m x n) matrix. As a consequence of the definition, the ith row of A is the vector representation of the oriented seg consisting of all of the edges incident to vertex Vi, with the orientation directed away from the vertex. It should not be difficult to see that the rank of A is given by the rank of the graph. Consequently, the rows of A span the space V,. Example. Consider the edge labeled i, indicates i ion and q=22=4.Addt tables : + 0 0 0 11032 2 2 3 3

weighted, directed graph shown in Fig. 1, where an that edge ei has weight wz’ = k. Also suppose that multiplication in GF(4) are given by the following 1

2

3

1

2

3

3 2

0 1

1 0

-

.

0123

00000 10123. 20231 30312

Choosing t = (b4, b,, b,) results in

Bj(G)=[

;

i

312200

Q,(G) =

012030 310001

352

8

;

;

i ],

BJG)=[

i

,1 = Q,(Q)

[;

i

8

i

i

i ],

;

i

;

K

;I.

Journal of The Franklin

Institute

Vector Spaces and the Four-Color

Problem

and 300230 A=

310001 012030’

Suppose that an n-edge graph has wi = 1 for i = 1,2, . . ., n. Then by setting q = 3, the above generalizations result in the standard ternary matrix descriptions of directed graphs. On the other hand, setting q = 2 results in the binary representations of the corresponding undirected graphs.

61

FIG.

1.

For the remainder of this paper, only the special case q = 4 is considered. Let G be any directed n-edge graph with wi = 1 for i = 1,2, . . ., n. Suppose that A is the incidence matrix over GF(4) of G. Due to the structure of GF(4), the directions of the edges of G are completely arbitrary and, hence, can be ignored. Consequently, the matrices B,(G), Q,(G) and A are identical to the corresponding matrices obtained for the modulo 2 (binary) case. Now consider V,, the row space over GF(4) of A. Suppose that G has m vertices, and suppose that the ith row of A is denoted by ri (; = 1,2, . . . . m). Since the rows of A span V,, any vector VE V, can be expressed as v = a,r,+a,r,+ . . . +a,r,,, where ai E GF(4) for i = 1,2, . . ., m. Now assume that the ith component of v is denoted by va and vi = OeGF(4). Since each column of A has exactly two nonzero entries, say in rows j and k, then aj = ak. Conversely, if vi # 0 ,then ai # ak. Therefore, the following theorem has been proved. Theorem I. The vertices of a graph are four-colorable (i.e. the vertices can be colored with four colors such that adjacent vertices do not have the same color) if and only if the vector space V, over GF(4) contains a vector which consists entirely of nonzero components. For Theorem I, the four colors correspond to the elements of GF(4). If every component of v = a, r1 + a2 rz + . . . +a,r,~ V, is nonzero, then vertex

Vol. 294, No. 5, November 1972

353

Leonard 8. Bobrow vi is colored aiE GF(4) for i = 1,2, . .., m. The coloring of a graph with the elements of GF(4) was first suggested by Tutte (2,3). As mentioned previously, for any graph G the null space of the row space of a fundamental circuit matrix B,(G) is V,. Suppose that the ith column of B,(G) is denoted by ui (i = 1,2,..., n). Then the following is an obvious corollary to Theorem I. Corollary 1. The vertices of a graph are four-colorable if and only if there are nonzero scalars a, E GF(4) for i = 1,2, . . . , n such that a,u,+aau,+

. . . +a,u,

= 0.

Now suppose that each edge of a graph G is colored with a nonzero scalar of GF(4). Then there is a one-to-one correspondence between all such colorings and all linear combinations of the form a, u1 + a2 u2 + . . . + a, un,, where ai # 0 for i = 1,2, . . . . n. In addition, each such linear combination of the columns of B,(G) consists of a set of m sums, each sum being associated with a fundamental circuit. From this point of view, Corollary 1 can be restated as the following theorem. Theorem II. The vertices of a graph G are four-colorable if and only if the edges of G can be colored with the nonzero elements of GF(4) such that the sum of the colors of the edges forming each fundamental circuit (and hence, every circuit) is 0 E GF(4). Theorem II generalizes a condition (4,5) that is both necessary and sufficient to four-color triangular graphs (i.e. planar graphs in which every region is bounded by exactly three edges). In developing Lemma 1, Theorem I, Corollary 1 and Theorem II, the concept of planarity was ignored. For the case of planar graphs, dual remarks can be made about coloring the regions of a graph. For example: The regions of a planar graph are four-colorable if and only if V, contains a vector which consists entirely of nonzero components. In addition : The regions of a planar graph are four-colorable if and only if the edges of G can be colored with 1, 2 and 3 such that the sum of the colors of the edges forming each fundamental seg is 0 EGF(~). This statement generalizes a condition (2,5,6) that is necessary and sufficient to four-color the regions of trivalent graphs (i.e. planar graphs in which every vertex has degree 3). References

(1) L. S. Bobrow and S. L. Hakimi, (2) (3) (4) (5) (6) (7)

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“Graph theoretic p-ary codes”, IEEE Tram. on Info. Theory, Vol. IT-17, No. 2, pp. 215-218, March 1971. W. T. Tutte, “On the imbedding of linear graphs in surfaces”, Proc. London Math. Sot., Ser. 2, Vol. 51, pp. 474483, 1949. W. T. Tutte, “On the algebraic theory of graph colorings”, J. Combin. Theoq, Vol. 1, No. 1, pp. 15-50, June 1966. C. R. Marathe, “On the dual of a trivalent Map”, Am. Math. Monthly, Vol. 68, No. 5, pp. 448-455, May 1961. R. G. Busacker and T. L. Saaty, “Finite Graphs and Networks: An Introduction with Applications”, New York, McGraw-Hill, 1965. C. Berge, “The Theory of Graphs”, New York, John Wiley, 1962. W. T. Tutte, “Even and odd 4-colorings”, in “Proof Techniques in Graph Theory”, ed. by F. Harary, pp. 161-169, New York, Academic Press, 1969.

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of The Franklin

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