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Statement

Notation as on the Theorem's page: an $r$-way joining two points is $r$ paths between them, pairwise sharing no line; $l_r(n)$ is the least number of lines that guarantees an $r$-way in a graph of $n$ points; a $J$-graph is a graph with $n$ points and $3n-3$ lines containing no 6-way.

Construction (p. 687, unnumbered). The paper calls bi-wheels the graphs of the types drawn in Figure 1 (p. 688, captioned "Bi-wheel blocks $J[n, 3n-3]$, for $n$ odd and even"), and asserts two things about them.

1. "Figure 1 establishes the existence of $J$-graphs $[n, 3n-3]$ for any $n$" (p. 687): for each $n$ there is a graph with $n$ points and $3n-3$ lines in which no two points are joined by a 6-way.
2. For any $r$, giving each outer-ring point of a bi-wheel more inner-ring neighbors yields a bi-wheel with $n$ points, $[r(n-1)/2]$ lines and no $r$-way, which the paper describes as "establishing a lower bound for $l_r(n)$" (p. 687). Since a graph with that many lines and no $r$-way exists, $l_r(n)\ge[r(n-1)/2]+1$.

The paper defines bi-wheels only by the figure and gives no proof of either assertion. It does not define the bracket; it is read here as the integer part, as in the corpus's other pages on this threshold. Neither assertion states a range of $n$, and neither can hold for every $n$: for $2\le n\le5$ a simple graph on $n$ points has at most $\binom n2<3n-3$ lines, so assertion 1 is meaningful for $n\ge6$ (where $n=6$ gives $K_6$, the paper's "$J[6,15]=K_6$", p. 688), and assertion 2 needs $[r(n-1)/2]\le\binom n2$, which for $n\ge3$ means $r\le n$.

Source. J. L. Leonard, Graphs with 6-ways, Canadian J. Math. 25 (1973), no. 4, 687--692: the assertions on p. 687 and Figure 1 on p. 688. The edition is identified in the source digest.

Read depth. Claims checked: the two assertions and the caption of Figure 1 were read clause by clause on the page images. The figure was not checked here to have $3n-3$ lines and no 6-way, and the general-$r$ construction was not checked; the paper supplies no argument for either. Nothing here is independently reviewed.

Proof pointer

None in the paper. Assertion 1 is what makes the Theorem's bound sharp, so that $l_6(n)=3n-2$; assertion 2 is stated without detail.

Dependencies

None.

Bears on

• Problem 915: assertion 2, read with the integer part, is the lower bound $\ell_m(n)\ge[m(n-1)/2]+1$ in the site's notation for the edge-disjoint reading. At the problem's parameters, $N$ copies with $1+(m-1)N$ vertices, it gives $\ell_m\ge\binom m2N+1$, the problem's edge count, the same bound the extremal example $K_1+NK_{m-1}$ gives (an arithmetic check made here). The paper asserts the construction without proof; Mader's Korollar states graphs with $[(n/2)(m-1)]$ edges on $m\ge n\ge2$ vertices and no $n$ edge-disjoint paths between any two vertices, in its own notation.