On the Line-Graph of the Complete Bigraph

Abstract

The line-graph, $L(G)$, of any given simple graph, $G$, is that graph whose points can be put in a one-to-one correspondence with the two edges of $G$ in such a way that two points in $L(G)$ are adjacent if, and only if, their corresponding edges in $G$ are adjacent. The complete $m$ by $n$ bigraph, $B_{mn}$, contains $mn$ edges which join each point in one set of $m$ points to each point in a second set, disjoint from the first, containing $n$ points. As general references on terminology see, e.g., Harary [4] and Ore [8]. If we suppose that $m \geqq n \geqq 1$ it can be seen that $L(B_{mn})$ has the following three properties: (1) The graph has $mn$ points each of which is adjacent to $m + n - 2$ other points. (2) $n \binom{m}{2}$ of the pairs of adjacent joints are mutually adjacent to $m - 2$ other points and the remaining $m \binom{n}{2}$ pairs of adjacent points are mutually adjacent to $n - 2$ other points. (03) Any two distinct nonadjacent points are mutually adjacent to two points. The object of this note is to show that if any graph satisfies these three conditions then it is isomorphic to $L(B_{mn})$ except possibly when $(m, n) = (4, 4), (4, 3)$ or $(5, 4)$. This will generalize a result of Shrikhande [10] (see also Mesner [7]) who, using different terminology, has already shown this for the case that $m - n$. The corresponding problem for the line-graph of the ordinary complete graph of $n$ points has been treated by Connor [3], Shrikhande [9], Hoffman [5], and Chang [1], [2].