On 4-Cycles and 5-Cycles in Regular Tournaments
On 4-Cycles and 5-Cycles in Regular Tournaments
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常规锦标赛中的 4 周期和 5 周期
DOI：
10.1112/blms/18.2.135
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发表时间：
1986
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影响因子：
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通讯作者：
P. Rowlinson
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文献类型：
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作者：
P. Rowlinson
首先，如果每个点具有indegree k，则有一些定义。它是不平凡的，t的任何两个点共同主导着t点，如果t是非平凡的，对于t的每个点V /。通过计算电视的弧线，我们看到V具有超级2T +1在+4处的偏斜式矩阵的存在。 QRP的点是字段ZP的P元素，当u-v是正方形时，u才是V。琐碎的和t的每个弧都位于同一数量的W-Cycles中如果这样的比赛有超过三个然后，它还具有属性 ^ 4和＆H，尽管具有属性或8PH的比赛一定是规律的，但仅仅是一个属性，以确保这一规律性。如下：它具有三个点偶会3-cycles CV C2，C3，Q的每个点占Q+ 1的每个点（其中订阅减少了模量3）。这场比赛的财产 ^4，但没有 ^3。 4和属性和h是双重的。
First, some definitions. A tournament is regular of degree k if each point has indegree k and outdegree k: clearly such a tournament has 2k +1 points. The trivial tournament has just one point. A tournament T is doubly regular with subdegree t if it is non-trivial and any two points of T jointly dominate precisely t points; equivalently if T is non-trivial and for each point v of T, the subtournament Tv on the points dominated by v is regular of degree /. By counting arcs in Tv we see that v has outdegree 2t +1 , and it follows that Tis regular of degree 2t+1. Reid and Brown [4] have shown that the existence of a doubly regular tournament with subdegree / is equivalent to the existence of a skew-Hadamard matrix of order At+4. The simplest examples of doubly regular tournaments are provided by the quadratic residue tournaments QRP, where p is a prime congruent to 3 modulo 4: the points of QRp are the p elements of the field Zp, and u dominates v if and only if u — v is a square inZp. Let m be an integer ^ 3. We say that a tournament T has property &m if T is non-trivial and each arc of T lies in the same non-zero number of w-cycles. It is well-known (see [4]) that a tournament has property ^ 3 if and only if it is doubly regular. Moreover, as we shall see, if such a tournament has more than three points then it also has properties ^4 and &h. Although a tournament with property ^ or 8Ph is necessarily regular, neither property alone is strong enough to ensure double regularity. To see this consider first the 9-point tournament Z3[Z3], defined as follows: it has three point-disjoint 3-cycles Cv C2, C3 and each point of Q dominates each point of Q+ 1 (where subscripts are reduced modulo 3). It is straightforward to check that this tournament has property ^4 but not ^3. Secondly, the 5-point tournament which is the union of two arc-disjoint 5-cycles has property ^ 5 but not ^3. We prove that a tournament which has both property ^4 and property &h is doubly regular. Our methods are algebraic and serve also to show that a regular tournament is doubly regular if and only if the minimal polynomial of its adjacency matrix has degree 3. 2. Some preliminary results