# Complete Graphs

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# Complete Graphs

In this section the complete graph on $k$ vertices, denoted $K_k$, is considered. A $K_k$-design of order $n$ is better known as a $(v,k,1)$-BIBD with $v=n$.

## Spectrum Results

For $k\leq 9$, Table 1 summarises the known results on the spectrum problem for $K_k$. An explanation of the sources of these results is given in .

Table 1 $k$ Spectrum for $K_k$ Possible exceptions $2$ ${\rm \ all \ }n$ $\emptyset$ $3$ $n \equiv 1 {\rm \ or \ } 3 \,({\rm mod \ }6)$ $\emptyset$ $4$ $n \equiv 1 {\rm \ or \ } 4 \,({\rm mod \ }12)$ $\emptyset$ $5$ $n \equiv 1 {\rm \ or \ } 5 \,({\rm mod \ }20)$ $\emptyset$ $6$ $n \equiv 1 {\rm \ or \ } 6 \,({\rm mod \ }15)$ ${\rm and \ }n \not\in\{16,21,36,46\}$ $n\in\{51,61,81,166,226,231,256,261,286,316,321,$ $346,351,376,406,411,436,441,471,501,561,591,616,$ $646,651,676,771,796,801\}$ $7$ $n \equiv 1 {\rm \ or \ } 7 \,({\rm mod \ }42)$ ${\rm and \ }n \neq 43$ $n=42t+1{\rm \ for \ }t\in\{2,3,5,6,12,14,17,19,22,27,33,$ $37,39,42,47,59,62\};{\rm \ and}$ $n=42t+7{\rm \ for \ }t\in\{3,19,34,39\}$ $8$ $n \equiv 1 {\rm \ or \ } 8 \,({\rm mod \ }56)$ $n=56t+1{\rm \ for \ }t\in\{2,3,4,5,6,7,14,19,20,21,22,24,$ $25,26,27,28,31,32,34,35,39,40,46,52,59,61,62,67\};$ ${\rm and}$ $n=56t+8{\rm \ for \ }t\in\{3,11,13,20,22,23,25,26,27,28\}$ $9$ $n \equiv 1 {\rm \ or \ } 9 \,({\rm mod \ }72)$ $n=72t+1{\rm \ for \ }t\in\{2,3,4,5,7,11,12,15,20,21,22,$ $24,27,31,32,34,37,38,40,42,43,45,47,50,52,53,\};$ $56,60,61,62,67,68,75,76,84,92,94,96,102,132,$ $174,191,194,196,201,204,209\}\};{\rm \ and}$ $n=72t+9{\rm \ for \ }t\in\{2,3,4,5,12,13,14,18,22,23,$ $25,26,27,28,31,33,34,38,40,41,43,46,47,52,59,$ $61,62,67,68,76,85,93,94,102,103,139,148,174,$ $183,192,202,203,209,229\}$

For $k\geq 10$, the known $K_k$-designs are given by Theorem 1 (and Wilson's Theorem).

Theorem 1

Let $p$ be a prime power and let $n,\, r$ and $s$ be integers such that $n \geq 2$ and $2 \leq r < s$.

• There exists a $K_q$-design of order $q^n$ (affine geometries) (see ).
• There exists a $K_{q+1}$-design of order $q^n + \ldots + q + 1$ (projective geometries) (see ).
• There exists a $K_{q+1}$-design of order $q^3+1$ (unital designs) (see ).
• There exists a $K_{2^{r-1}}$-design of order $2^{r-1}(2^r-1)$ (oval designs) .
• There exists a $K_{2^r}$-design of order $2^{r+s}+2^r-2^s$ (Denniston designs) .

Obvious necessary conditions for the existence of a $K_k$-design of order $n$ are

• $n = 1$ or $n \geq k$;
• $n(n-1)\equiv 0 \,({\rm mod \ }k(k-1))$; and
• $n \equiv 1 \,({\rm mod \ }k-1))$.

Further necessary conditions for the existence of a $K_k$-design of order $n$ are described in the following three theorems.

Theorem 2 

There is no $K_k$-design of order $n$ for any $n$ in the range $k < n < k^2-k+1$.

Theorem 3 , 

If $k \equiv 2 {\rm \ or \ } 3 \,({\rm mod \ }4)$ and $k-1$ is not the sum of two integer squares then there is no $K_k$-design of order $k^2-k+1$ and no $K_{k-1}$-design of order $k^2-2k+1$.

Theorem 4 

There is no $K_{10}$-design of order $100$ and no $K_{11}$-design of order $111$.

## Notes

• A $K_3$-design is a Steiner triple system.

## References

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