(Edge Coloring of graph

Question

The main I want to color all edge off a complete graph where 1-color triangle (monochromatic that mean all edges of triangle has same color ) and $3$-color triangle ( mean all edges of triangle color by different colors). I suppose, having a blue triangle and I rotate a blue triangle $7$ times does not matter the direction of rotation either clockwise or not, so I get $(v_1,v_2,v_6)$ connect by a blue and $(v_2,v_3,v_7)$ by red , $(v_3,v_4,v_1)$ by green, $(v_5,v_4,v_2)$ by yellow, $(v_6,v_5,v_3)$ by orange , $(v_7,v_6,v_4)$ by black and $(v_1,v_7,v_5)$ by brown.

I am unable to write the method mathematically as function, series, or algorithm to generalize that for $n$ . Some help please or mention some link could lead me !

Answer 1

I guess you are asking about a coloring of edges of a complete graph $K_n$ such that each triangle (that is, a copy of $K_3$) have edges either the same color or distinct colors. It can be trivially provided by coloring each edge in a unique color.

Not so trivial example is shown at your picture, where is presented a partition of edges of $K_7$ into triangles. Such a partition of $K_n$ also provides a valid coloring. The question when $K_n$ admits such a partition is well-known. In fact, this is a question whether there exists a Steiner triple system $S(2,3,n)$. Since the graph $K_n$ has ${\binom{n}{2}=n(n-1)/2}$ edges and each triangle has three edges, a necessary condition is $6|n(n-1)$. Morever, since each vertex should have even degree, $n$ is odd. This follows that $n\equiv 1\pmod 6$ or $n\equiv 3\pmod 6$. This condition is also sufficient for the existence of a Steiner system $S(2,3,n)$, that was shown by Kirkman in 1847 (see, for instance, [Bol, p.113]).

References

[Bol] B. Bollobás, Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability, Cambridge University Press, 1st edition, 1986.