On the Line Graph of the Complete Bipartite Graph
In an interesting recent article [4], J. W. Moon has given a list of properties of the graph $L(B_{mn})$ (to be defined more precisely below) and investigated the question of whether these properties characterize the graph. In case $m = n$, this question had been settled by Shrikhande [5] (see also [1], [2] and [3]), who proved that the answer was yes unless $m = n = 4$, when there is exactly one exception. In case $m > n$, Moon shows that the answer is yes unless $(m, n) = (5, 4)$ or (4, 3), which cases were left unsettled in [4]. The purpose of this note is to show that, in those cases as well, the answer is yes, thus completing Moon's discussion. Now to define our problem more exactly. The line graph of the complete bipartite graph on sets with $m$ and $n$ vertices, denoted by $L(B_{mn})$, is the graph with $mn$ vertices given by all ordered pairs $(i, j), 1 \leqq i \leqq m, 1 \leqq j \leqq n$. Two vertices $(i, j)$ and $(i', j')$ are joined by an edge if $i = i'$ or $j = j'$, but not both. The graph $L(B_{mn})$ has the following properties: (1.1) It has $mn$ vertices. (1.2) Each vertex has valence $m + n - 2$. (1.3) If two vertices are not adjacent, there are exactly two vertices adjacent to each. (1.4) Of the $\frac{1}{2} mn(m + n - 2)$ pairs of adjacent vertices, exactly $n\binom{m}{2}$ pairs are each adjacent to exactly $m - 2$ vertices, the remaining $m\binom{n}{2}$ pairs are each adjacent to exactly $n - 2$ vertices. We now assume $m > n$, and let $G_{mn}$ be any graph satisfying (1.1)-(1.4). Our object is to prove that when $(m, n) = (5, 4)$ or (4, 3), $G_{mn} = L(B_{mn})$. Moon has established $G_{mn} = L(B_{mn})$ in all other cases.