Rainbow Graphs and Switching Classes

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1 Rainbow Graphs and Switching Classes The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Oh, Suho, Hwanchul Yoo, and Taedong Yun. Rainbow Graphs and Switching Classes. SIAM Journal on Discrete Mathematics 7, no. (April 4, 0): , Society for Industrial and Applied Mathematics Society for Industrial and Applied Mathematics Version Final published version Accessed Tue Jan 04:6:08 EST 09 Citable Link Terms of Use Detailed Terms Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.

2 SIAM J. DISCRETE MATH. Vol. 7, No., pp. 06 c 0 Society for Industrial and Applied Mathematics RAINBOW GRAPHS AND SWITCHING CLASSES Downloaded 09/09/ to Redistribution subject to SIAM license or copyright; see SUHO OH, HWANCHUL YOO, AND TAEDONG YUN Abstract. A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection between the isomorphism classes of n-rainbow graphs on n vertices and the switching classes of graphs on n vertices. Key words. rainbow graph, switching class, Seidel matrix, Eulerian graph, two graph, parallelotopic graph, neighbor-complete coloring AMS subject classifications. 05A9, 05C0, 05E8 DOI. 0.7/ Introduction. Throughout this paper, we will use the term graph to refer to a graph on unlabeled vertices without loops or multiple edges, and labeled graph to refer to a graph on labeled vertices without loops or multiple edges. A vertex-coloring (with n colors) of a graph is called an n-rainbow coloring if every color appears once, and only once, in each neighborhood of a vertex. Note that an n-rainbow coloring is not a proper coloring. A graph is called an n-rainbow graph if the graph admits an n-rainbow coloring. This term was initially coined by Woldar [7], but the same concept appeared earlier in Ustimenko [6] and Lazebnik and Woldar [] under the names parallelotopic graph and neighbor-complete coloring, respectively. There are interesting instances of rainbow graphs appearing throughout mathematics and theoretical computer science literature. For example, one can naturally construct a rainbow graph from a group [7] or from a system of equations [], and the structure of such rainbow graphs has been shown to be closely related to the structure of the underlying algebraic structure. We state here without proof some general facts about rainbow graphs, which are all easy to verify. Many of these facts can be found in [7].. An n-rainbow graph is n-regular.. Each color occurs equally often in the vertex set of an n-rainbow graph.. An n-rainbow graph contains a perfect matching, which is constructed by choosing edges whose endpoints have the same color. 4. The number of vertices of an n-rainbow graph is a multiple of n. On the other hand, switching avertexv of a labeled graph G means reversing the adjacency of v and w for every other vertex w: Ifv and w are connected, then we delete the edge; otherwise we add an edge between them. This operation was defined by Seidel [5]. Two labeled graphs are switching equivalent if one can be obtained from the other by a sequence of switching operations. Two unlabeled graphs are switching Received by the editors November 4, 0; accepted for publication (in revised form) March 8, 0; published electronically June 9, 0. Department of Mathematics, University of Michigan, Ann Arbor, MI 4809 (suhooh@umich.edu). This author s work was partially supported by a Samsung Scholarship. Department of Mathematics, Korea Institute for Advanced Study, Seoul 0-7, Republic of Korea (hcyoo@kias.re.kr). This author s work was partially supported by the National Research Foundation of Korea. Department of Mathematics, MIT, Cambridge, MA 09 (tedyun@math.mit.edu). This author s work was partially supported by a Samsung Scholarship. 06

3 RAINBOW GRAPHS AND SWITCHING CLASSES 07 Downloaded 09/09/ to Redistribution subject to SIAM license or copyright; see equivalent if we can label the vertices so that they become switching equivalent. The equivalent classes coming from this equivalence relation are called switching classes. Mallows and Sloane [4] and Cameron [] showed that the number of switching classes equals the number of unlabeled Eulerian graphs. The main result of this paper is the following theorem. Theorem.. The number of isomorphism classes of n-rainbow graphs with n vertices is the same as the number of switching classes of graphs with n vertices. Corollary.. The number of isomorphism classes of n-rainbow graphs with n vertices is the same as the number of unlabeled Eulerian graphs (every vertex has even degree) with n vertices. We will explicitly construct a bijective map between the classes.. Map from switching classes to n-rainbow graphs. In this section, we introduce a map that sends a switching class of graphs having n vertices to an isomorphism class of n-rainbow graphs having n vertices. This map will be shown to be a bijection in section. Choose a graph G =(V,E) thathasn vertices without loops and multiple edges. Label the vertices from to n. Now let G =(Ṽ,Ẽ) be a graph having n vertices such that. the vertices are labeled by {,...,n,,...,n },. (i, i ) Ẽ,. (i, j), (i,j ) Ẽ if (i, j) E, 4. (i,j),(i, j ) Ẽ if (i, j) / E. Then the underlying unlabeled graph of G is n-rainbow because we can color the vertices labeled by i and i with the color i, and this is clearly a rainbow coloring. An example is given in Figure. We denote by [G] the switching class of G. Now let us define the desired map ψ by sending [G] to the underlying unlabeled graph of G. Toseethatψ is well defined, let H be the graph obtained from G by switching at a vertex labeled by i. Let H be the graph with n vertices obtained from H by the previous method. Now if we interchange the labels of i, i in H, we recover G. An example of this phenomenon is given in Figure. Proposition.. ψ is surjective. Proof. For any n-rainbow graph G on n vertices, assign an n-rainbow coloring on its vertex set. Now let G be an induced subgraph of G having n vertices on which each color appears once. Clearly, ψ([g]) = G. Remark.. A similar map was constructed in [], with a different coloring of graphs, to prove that the switching equivalence problem is polynomial time equivalent to the graph isomorphism problem.. The main proof. In this section, we will prove our main result, Theorem.. To do so, let us recall the definition of Seidel matrices, originally defined in [5]. Definition.. Let G be a graph over n vertices. The Seidel matrix of G is a symmetric n n matrix with entries a ij such that a ii =0for all i [n], a ij = a ji =if (i, j) E(G), a ij = a ji = if (i, j) / E(G).

4 08 SUHO OH, HWANCHUL YOO, AND TAEDONG YUN Downloaded 09/09/ to Redistribution subject to SIAM license or copyright; see Fig.. The map that sends G to G. Fig.. The exchange equivalence. Definition.. For a Seidel matrix A we define à as ( ) A I A à :=. I A A Note that à is also a Seidel matrix. In particular, if A is the Seidel matrix of G, then à is the Seidel matrix of G. Let A and B be Seidel matrices of some graphs G and G, respectively. Then G and G are isomorphic if and only if B = PAP for some permutation matrix P. Recall that the signed permutation matrices are the matrices we get from permutation matrices by replacing s with ± s. Since switching at vertex i of a graph corresponds to changing signs of entries in the ith row and the ith column of its Seidel matrix, G and G are switching equivalent if and only if B = QAQ for some signed permutation matrix Q. In order to prove Theorem., it remains only to prove that ψ is injective, since surjectivity is given by Proposition.. We need to show that if G is isomorphic to G,thenG and G are in the same switching class. This can be restated in the following linear algebra language. Proposition.. If two Seidel matrices A and B satisfy à = P BP,where P is a n n permutation matrix, then there exists an n n signed permutation matrix Q such that A = QBQ. We prove this proposition by a sequence of lemmas given below. Lemma.4. Let P = ( P P ) P P 4 be a n n permutation matrix, where P,...,P 4 are n n matrices. The following equations hold: () P P T =0, P P4 T =0, P P T =0, P 4P T =0. () P P T + P P T = I, P P T + P 4 P4 T = I.

5 RAINBOW GRAPHS AND SWITCHING CLASSES 09 Downloaded 09/09/ to Redistribution subject to SIAM license or copyright; see Proof. FromPP T = I we have ( P P T + P P T P P T + P P4 T ) ( ) I 0 P P T + P 4P T P P T + P 4P4 T =. 0 I The claim follows from the fact that every entry of each P i is nonnegative. Lemma.5. In the assumption of Proposition. and Lemma.4, letz denote (P P P + P 4 )/. Then, we have A I = Z(B I)Z T. Proof. Ã = P BP implies ( ) ( )( )( A I A P P = B I B P T P T I A A P P 4 I B B P T P4 T It follows from Lemma.4 that A = XBX T + I XX T, A = XBY T + I XY T, A = YBY T + I YY T, A = YBX T + I YX T, where X = P P and Y = P 4 P. Summing these four equations, we get Hence, 4A =(X + Y )B(X + Y ) T +4I (X + Y )(X + Y ) T. ( X + Y A I = ) ( ) T X + Y (B I). Definition.6. Let S be an m m matrix. We call S a signed half-permutation matrix if every row and column of S has either () exactly one nonzero entry α with α = ±,or () exactly two nonzero entries α and β with α = ±, β = ±. For instance, a matrix is a signed half-permutation matrix. Definition.7. Let S be an m m signed half-permutation matrix. We say that an m m matrix T is an integration of S if the following conditions hold for all i =,...,m: ).

6 0 SUHO OH, HWANCHUL YOO, AND TAEDONG YUN Downloaded 09/09/ to Redistribution subject to SIAM license or copyright; see () If the ith row (resp., column) of S has only one nonzero entry (which is ±), then the ith row (resp., column) of T is the same as the ith row (resp., column) of S. () If the ith row (resp., column) of S has two nonzero entries so that it is of form [ ( )p ( )q ], then the ith row (resp., column) of T is obtained by doubling one of the nonzero entries and setting the other entry to zero. In other words, the ith row (resp., column) of T should look like either or [ ( ) p 0 ] [ 0 ( ) q ]. Clearly an integration of a signed half-permutation matrix is a signed permutation matrix. For instance, the matrix is an integration of the signed half-permutation matrix Lemma.8. Every signed half-permutation matrix has an integration. Proof. We can find an integration T of the signed half-permutation matrix S by the following method. First, pick an entry α of S with value α = ± in S. Replace α by sgn(α ), where sgn(α) isifα is positive and is ifα is negative. Let α = ± be the other nonzero entry in the same row as α. Replace α by 0. Let α = ± be the other nonzero entry in the same column as α. Replace α by sgn(α ). By continuing this process, alternating row and column, we will end up at α again since there are only a finite number of entries. If there is no entry with value ±, then we are done. Otherwise, we repeat this process until there is no entry with value ±. Let T be the matrix we get after performing this algorithm to S. Then, T satisfies the two conditions of Definition.7. Lemma.9. Let M, N be m m matrices such that each entry m ij of M satisfies m ij and every entry of N is either or. If MZ = N for some signed half-permutation matrix Z, then every entry of M is either or. Furthermore,if Q is an integration of Z, then we have MQ = N. Proof. Ifthejth column of Z has an entry z ij = ±, then m ki = ±n kj = ± for all k =,...,m. Otherwise, it has two nonzero entries, z ij = ± and z i j = ±,

7 RAINBOW GRAPHS AND SWITCHING CLASSES Downloaded 09/09/ to Redistribution subject to SIAM license or copyright; see for some i and i.inthiscase, m ki z ij + m ki z ij, and the equalities are satisfied only if m ki =n kj z i and m ki =n kj z i. Hence every entry of M is either or. Moreover, if Q is an integration of Z, then it is a signed permutation matrix with q ij = z ij if z ij = ± andq ij =z ij or 0 if z ij = ±. Therefore m kiq ij = n kj, and we have MQ = N. Now we are ready to prove Proposition., which finishes the proof of Theorem.. Proof of Proposition.. By Lemma.5, A I = Z(B I)Z T,whereZ = (P P P + P 4 )/. We can see that Z is a signed half-permutation matrix. Let Q be an integration of Z. Obviously Q T is an integration of Z T. Since A, B are Seidel matrices, every entry of A I is either or, and every entry of Z(B I) is between and. By Lemma.9, every entry of Z(B I) iseitheror, and A I = Z(B I)Q T. Using Lemma.9 again on the transpose of Z(B I), we get Z(B I) =Q(B I). Therefore A I = Q(B I)Q T = QBQ T I, and hence A = QBQ T. Acknowledgments. We would like to thank Tanya Khovanova for introducing the following puzzle that led us to the study of rainbow graphs: A sultan decides to check just how wise his two wise men really are. The sultan chooses a cell on a chessboard and shows it only to the first wise man. In addition, each cell on the chessboard either contains a pebble or is empty. Then, the first wise man either removes a pebble from a cell that contains a pebble or adds a pebble to an empty cell. Next, the second wise man must look at the board and guess which cell was chosen by the sultan. The two wise men are permitted to agree on the strategy beforehand. The strategy for ensuring that the second wise man will always guess the chosen cell comes from assigning a rainbow coloring on a 64-regular graph. We also thank the anonymous referees for pointing us to references [] and [6]. REFERENCES [] P. Cameron, Cohomological aspects of two-graphs, Math. Z., 57 (977), pp [] C. Colbourn and D. Corneil, On deciding switching equivalence of graphs, Discrete Appl. Math., (980), pp [] F. Lazebnik and A. Woldar, General properties of some families of graphs defined by systems of equations, J. Graph Theory, 8 (00), pp [4] C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equal in number, SIAM J. Appl. Math., 8 (975), pp [5] J. Seidel, A survey of two-graphs, in Colloquio Internazionale Sulle Teorie Combinatorie (Rome, 97), Tomo I, Atti dei Convegni Lincei 7, Accad. Naz. Lincei, Rome, 976, pp [6] V. Ustimenko, Coordinates of trees and their quotients, in Voronoi s Impact on Modern Science, Vol., Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, 998, pp [7] A. Woldar, Rainbow graphs, in Codes and Designs (Columbus, OH, 000), Ohio State Univ. Math. Res. Inst. Publ. 0, de Gruyter, Berlin, 00, pp..