Almost-rainbow edge-colorings of some small subgraphs

Let $f(n,p,q)$ be the minimum number of colors necessary to color the edges of $K_n$ so that every $K_p$ is at least $q$-colored. We improve current bounds on the {7/4}n-3$, slightly improving the bound of Axenovich. We make small improvements on bounds of Erd\H os and Gy\'arf\'as by showing${5/6}n+1\leq f(n,4,5)$and for all even$n\not\equiv 1 \pmod 3$,$f(n,4,5)\leq n-1$. For a complete bipartite graph$G=K_{n,n}$, we show an n-color construction to color the edges of$G$so that every$C_4\subseteq G$ is colored by at least three colors. This improves the best known upper bound of M. Axenovich, Z. F\"uredi, and D. Mubayi.Comment: 13 page

All trees are six-cordial

For any integer $k>0$, a tree $T$ is $k$-cordial if there exists a labeling of the vertices of $T$ by $\mathbb{Z}_k$, inducing a labeling on the edges with edge-weights found by summing the labels on vertices incident to a given edge modulo $k$ so that each label appears on at most one more vertex than any other and each edge-weight appears on at most one more edge than any other. We prove that all trees are six-cordial by an adjustment of the test proposed by Hovey (1991) to show all trees are $k$-cordial.Comment: 16 pages, 12 figure