Solutions for two conjectures on kaleidoscopic edge-colorings

TL;DR

This paper investigates kaleidoscopic edge-colorings in graphs, determining the minimal number of colors for complete graphs to be kaleidoscopes and constructing regular kaleidoscopes of specific orders, thus resolving two conjectures.

Contribution

It precisely determines the minimal number of colors needed for complete graphs to be kaleidoscopes and constructs regular kaleidoscopes of certain orders, solving two open conjectures.

Findings

Complete graphs $K_n$ are $k$-kaleidoscopes for specific $k$, confirming a conjecture.

Constructed $r$-regular 3-kaleidoscopes of order $inom{r-1}{2}-1$, solving a conjecture.

Established the minimal $k$ for complete graphs to be kaleidoscopes.

Abstract

For an $r$-regular graph $G$, we define an edge-coloring $c$ with colors from $\{1,2,\cdots,$ $k\}$, in such a way that any vertex of $G$ is incident to at least one edge of each color. The multiset-color $c_m(v)$ of a vertex $v$ is defined as the ordered tuple $(a_1,a_2,\cdots ,a_k)$, where $a_i \ (1\leq i\leq k)$ denotes the number of edges with color $i$ which are incident with $v$ in $G$. Then this edge-coloring $c$ is called a {\it $k$-kaleidoscopic coloring} of $G$ if every two distinct vertices in $G$ have different multiset-colors and in this way the graph $G$ is defined as a {\it $k$-kaleidoscope}. In this paper, we determine the integer $k$ for a complete graph $K_n$ to be a $k$-kaleidoscope, and hence solve a conjecture in [P. Zhang, A Kaleidoscopic View of Graph Colorings, Springer, New York, 2016] that for any integers $n$ and $k$ with $n\geq k+3 \geq 6$, the complete graph $K_n$ is a $k$-kaleidoscope. Then, we construct an $r$-regular $3$-kaleidoscope of order $\binom{r-1}{2}-1$ for each integer $r\geq 7$, where $r\equiv 3\ (\text{mod}\ 4)$, which solves another conjecture in the same book on the maximum order for $r$-regular $3$-kaleidoscopes.