On Swell – Colored Complete Graphs 305

An edge-colored graph is said to be swell-colored if each triangle contains exactly 1 or 3 colors but never 2 colors and if the graph contains more than one color. It is shown that a swell-colored complete graph with n vertices contains at least √ n + 1 colors. The complete graph with n 2 vertices has a swell coloring using n + 1 colors if and only if there exists a finite affine plane of order n. A graph with its edges colored is said to be well-colored if each triangle contains exactly 1 or 3 colors but never 2 colors. Since all graphs can be well-colored using exactly one color, those graphs which are well-colored with more than one color will be referred to as swell-colored graphs or swell graphs for short. We shall investigate the number of colors with which a complete graph can be swell-colored. The complete graph on n vertices (generically denoted K n) can never be swell-colored with exactly two colors. A simple investigation shows that K 3 and K 4 are the only K n swell-colorable with exactly 3 colors; the other K n require more colors since they are more highly connected. For a particular value of n, what is the fewest number of colors that can give a swell K n ? This minimum completely characterizes the possible number of colors found in other swell-colorings of K n : Proposition 1. If the complete graph on n vertices can be swell-colored using exactly ρ colors, ρ < n 2 , then it can be swell-colored using exactly ρ + 1 colors. Before we prove this, we shall specify some terms and notation. Definition. A color-component of edge-colored graph G is a maximally connected subgraph (with edge colorings inherited from G) whose edges are all of the same color. If a color-component of G has edges of color c, then we call it a c-component of G. If G is complete, then every two vertices v 1 , v 2 are contained in a color-component, which we denote ←→ v 1 v 2. This is to be distinguished from v 1 v 2 which denotes the edge connecting v 1 and v 2 .