JOURNAL OF COMBINATORIAL THEORY, Series B 45, 115-l 19 (1988)

Note

Amalgamation of Team Tournaments*

H. LAKSER AND N. S. MENDELSOHN

Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

AND

J. W. MOON

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Communicated by W. T. Tutte

Received June 30, 1983

A team tournament consists of teams of players all of different strengths. Each

pair of teams competes by having each player of one team play each player of the other team, and the results of the competition yield an oriented graph. We prove

that, given three oriented graph structures G,, G2, and G on a finite set of vertices, there are disjoint representations of G, and Gz by team tournaments such that com-

bining the teams for Gr and Gz corresponding to each vertex yields a representation of G. 0 1988 Academic Press, Inc.

Let Y denote an integer-valued function defined for 1 < i,j < n such that r(i,j) = -r(j, i). We call r a weighting of the (arcs of the) oriented graph G with vertices the integers 1, . . . . n if r(i, j) > 0 iff the arc (i, j) is in G; we let (G, r) denote the graph G with the weighting r.

If T = (T,, . . . . T,) is a list of n teams of players, let cr(i,j) denote the num- ber of matches won by a player from Ti minus the number of matches won by a player from T, when each player from Ti plays one match against each player from T,. We identify players with numbers representing their strength; we assume that no two players have the same strength and that the stronger player always wins. We say that the collection T generates (G, r) or that E(T) = (G, r) iff a(i,j) = r(i,j) for 16 i,j<n.

* The research of all three authors was supported by grants from the Natural Sciences and Engineering Research Council of Canada.

11.5 009%8956/88 $3.00

582b/45/1-8 Copyright 0 1988 by Academic Press, Inc.

All rights of reproductiorl in any form reserved.

116 LAKSER, MENDELSOHN, AND MOON

Suppose T, = (T,, , . . . . T,,) and T, = (T,,, . . . . a,,) are two lists of teams such that no two players involved have the same strength. If T= (T,, . . . . T,,), where Tj= r,, u Tlz, we write T=T,@T2 and say that T is the amalgamation of T, and T,.

In [ 11, Moon and Moser gave an example of the two-team tournament T= ((2, 3, lo>, { 1, 8,9}) with cr(l,2) = 1 such that amalgamating with ({5}, j4)) results in a( 1, 2) =0 and amalgamating further with ({7}, (6)) results in ‘x( 1, 2) = - 1, that is, the paradoxical result that successively adding stronger players to the stronger team weakens it. One of the results of our present paper is that the general analogous situation holds-given any three oriented graphs G,, G,, G on the vertices 1, . . . . n there are T,, T, with Gi the underlying graph of Ti and with G the underlying graph of T, OT,.

LEMMA 1. Let (G, r) denote a weighted graph such that r(i, j) is evenfor I d i, j < n; then there is a list of teams T =‘( T,, . . . . T,) with al(T) = (G, r).

Proof: If all the weights r(i,j) are 0 let the teams be

{I, A}, (2, -2}, . ..) (n, -?z}.

Suppose a list of teams has been defined; let N denote a positive integer that exceeds the absolute value of the strength of any player in these teams. If the ith and jth teams are augmented by {N+ n, -N-n + 1) and {N+n-1, -N-n), respectively, and the remaining teams are augmented by

{N+l, -N-l}, (N+2, -N-2} ,..., (N+n-2, -N-n+2),

then the only effect this has on x is to increase cr(&j) and -n(j, i) by 2. Any graph (G, r) with even weights can be generated by repeating this

procedure as often as necessary.

We say that a function p defined on the vertices of G is a parity function if p(i) equals 0 or 1 for each vertex i. We say that r is a proper weighting of G if there is a parity function p such that

r(i,j)=p(i)p(j) (mod 2) (1)

for all i<j. Clearly, if r(i,j) is ever odd then r is a proper weighting iff the subgraph of G induced by those arcs (i, j ) for which r(i, j) is odd is a corn-, plete subgraph; it follows that p(i) = 1 for the vertices in this subgraph and p(i) = 0 for the remaining vertices. If r(i, j) is always even then r is certainly a proper weighting and p(i) = 1 for at most one vertex i. We shall thus henceforth work under the convention that if a proper weighting r is

AMALGAMATION OF TEAM TOURNAMENTS 117

specified in which all weights are even, then it is also specified whether p(i) ever equals 1 and, if so, for which vertex i. Thus, with this convention, if r is any proper weighting of G there is a unique parity function p that satisfies (1).

LEMMA 2. Let (G, r) be a weighted graph; then there exists a list of teams T = (T,, . . . . T,,) such that a(T) = (G, r) if and only ij” r is a proper weighting.

Proof: If (G, r) = cc(T), then r(i,j) E lTil . IT,1 (mod 2). The numbers I T,I reduced module 2 define a parity function p on the vertices of G; so r is a proper weighting of G.

Assume now that r is a proper weighting whose associated parity function is p. If p(i) = 0 for all vertices i, then all weights r(i, j) are even and the result follows from Lemma 1. We may therefore assume, in view of the observation following (1 ), that p(i ) = 1 iff 1~ i < u for some integer U.

Consider the oriented graph G’ obtained by removing all arcs (i,j) from G such that r(i,j) = 1 and 16 i <j < U. Define the weighting r’ of G’ by setting r’( i,j) = r(i,j) except that if 1 6 i <j< u then r’(i, j) = r(i, j) - 1 and r’(j, i) = r(j, i) + 1. Since r(i,j) is odd iff 1 < i <j < U, it follows that all the weights r’(i, j) are even, and so, by Lemma 1, that there exists a list of teams T’ = (T; , . . . . 7’:) with cr(T’) = (G’, r’). We may assume that each team Ti has an even number of players, half with positive strengths and half with negative strengths. Since adding any positive constant c to all the positive strengths does not affect cx(i,j) for T’, we may further assume that there are no players of strength 1, 2, . . . . u on the teams of T’. Let Ti= Ti u {u+l-i} for 1 <i<u and T;=Ti for u<i<n. Let T=(T,,..., T,); then a(T) = (G, r), as required.

Let Gi, G,, G be three oriented graphs defined on the vertices { 1, . . . . n}. Let r, and r2 denote proper weightings of the respective graphs G, , G, with associated parity functions p1 and p2. If r denotes any weighting of G we say that r satisfies the parity condition with respect to rl and rz (which, of course, determine G, and G2) if

r(i,j) = (PI(i) +P2(i))(Pl(j) +P2(j)) (mod 2)

for all 1 <i,j<n. Now

PI(i +fGi)PAj) -rl(4A + r2(igj) (mod 2),

by (1); so an equivalent formulation of the parity condition is that

r(i,j) - rl(i,j) - rz(i,j) -pl(i)p2(j) +Pl(.dP2(i) (mod 2)

for all l<i,j<n.

(2)

(3)

118 LAKSER, MENDELSOHN, AND MOON

THEOREM. Let G,, GZ, G be three oriented graphs on the set of vertices { 1, ..., n}. Let ri denote a proper weighting of Gi, i= 1,2. If r denotes any weighting of G, then there exist two lists of teams T, and T, such that 4Tl) = (G,, rl), @J= (G,, rz), and a(T, @T,) = (G, r) if and only if r satisfies the parity condition with respect to r, and r2.

ProoJ: Let pI and pz denote the parity functions associated with the weightings rl and ra. Suppose there exist two lists of teams T, = (T,,, . . . . T,,) and T, = (T12, . . . . TEz) such that cr(T,) = (G, rl), ct(T2)= (G, rz), and a(T, @T,) = (G, r), where we assume that no two players involved in the teams have the same strength. Then

r(i,j) = ~(T,I LJ Tiz, Tj, U Tj,)

-~T~L~T~~IT’I~T’~I

= (PI(i) +Pz(i))(Pl(j) +P,w) (mod 2)

for 1 < i, j < n; so r satisfies the parity condition with respect to rI and r2. To prove the converse, let r satisfy the parity condition with respect to r1

and r2. Consider an oriented graph H with 2n vertices x1, . . . . x, and Y , , . . . . y,. Define a weighting s of H as follows: if 1 < i < n, then

St-x,, Vi)= -s(Yi2 Xi) =pl(i)p2(i); (4)

and if 1 <i<j<n, then

s(x;, xj) = -s(xj, xi) = r,(i, j), (5)

f(Yi, Yj) = -s(Yj3 Yl) = rdi,j), (6)

s(Xj, Y,) = -dYf, xj) = -P,(j) PAi), (7)

s(xi, Y,) = -.$y;, xi) = r(i,j) - rl(i,j) - r2(i,j) -PI(j) p2(i). (8)

Define a parity function p for H by setting

We claim for all pairs v and w of H that

s(fA WI =p(v) P(W) (mod 2). (9)

If (v, w} equals {xi, y,} or {xi, yi} for i <j then (9) follows from (4) and (7). If {v, w} equals {xi, .x~> or (y,, yj} then (9) follows from (5), (6), and the fact that rl and rz are proper weightings of G with parity functions p,

AMALGAMATION OF TEAMTOURNAMENTS 119

and p2. Finally, if (0, w} = {xi, y,} with i<j then the parity condition (3) and formula (8) imply (9). It thus follows that s is a proper weighting of H.

Thus, by Lemma 2 applied to (H, s), there exist lists of teams T, = CT,, > . . . . T,,) and T, = (T,,, . . . . Tn2) such that no two players involved have the same strengths and such that if team T,, is associated with vertex xi and team Ti2 is associated with vertex yi then a( T,,, . . . . T,,, , T12, . . . . T,,) = (ff, s). By (5) and (6), a(T,) = (G, rl) and a(T2) = (G, r2). Finally, for each i, let T, = T,, u Ti2. Then

x(Ti, Tj) = a(Ti, U Ti2, T,, U Tj2)

=u(Til, T’,)+a(Ti2, T’2)+x(Tj1, Tj2)+ct(Ti2, Tj,)

= s(xiY xj) + s(Yi, Yj) + dx2, Yj) + dYi3 xj)

= r,(i, j) + r2(i, j) + r(i, j) - r,(i, j) - r,(i, j)

-pl(j)p2(i) +Pl(j)p2(i)

= r(i, j)

whenever i< j by (5), (6), (7), and (8). That is, cz(T, OT,) = (G, Y), com- pleting the proof of the theorem.

Since any oriented graph has an even weighting which is trivially proper and, equally trivially, satisfies the parity condition (2), we have the follow- ing corollary.

COROLLARY. Let G,,G,, and G be oriented graphs on the vertices 1, . . . . n. Then there exist two lists of teams T, and T, with each Gi the underlying graph of a(T,) such that G is the underlying graph of a(T, @T,).

The theorem and its corollary are an abstract representation theorem. We close this paper by presenting the following conjecture concerning the concrete representation problem.

Conjecture. Let Gi, G,, G be three oriented graphs on the set of vertices { 1, . . . . n>. Let ri denote a proper weighting of Gj, i=: 1,2. Let r be a weighting of G that satisfies the parity condition with respect to r1 and r2.

Let T, be a list of teams with a(T,)= (G,, rl). Then there exists a list of teams T2 with a(T,) = (G,, r2) such that cl(T, OT,) = (G, r).

REFERENCE

1. J. W. MORN AND L. MOSER, Generating oriented graphs by means of team comparisons,

Pacific J. Math. 21 (1967), 531-535.