Edge colorings of \(K_{2n}\) with a prescribed condition. I (Q1971213)

A graph \(L\) is called a lantern if it has two adjacent vertices \(u\), \(v\) such that all the other vertices of \(L\) are adjacent to both \(u\) and \(v\), and \(L\) has no other edges. The main result of this paper is the following: If \(L\) is a lantern of order \(2n\geq 8\), then any edge-coloring of \(L\) using \(2n-1\) colors can be extended to a proper edge-coloring of \(K_{2n}\) using the same set of colors. As a consequence, it is shown that if \(P\) is a partial symmetric latin square of order \(2n\geq 8\) such that two rows and two columns of \(P\) are full and all the other cells are empty and the two preassigned symbols in the main diagonal of \(P\) are the same, then \(P\) can be completed to form a symmetric latin square of order \(2n\) with constant diagonal.