On cyclically-interval edge colorings of trees

For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $\varphi:E(G)\rightarrow{1,2,\ldots,t}$ is called a proper edge $t$-coloring of a graph $G$ if adjacent edges are colored differently and each of $t$ colors is used. An arbitrary nonempty subset of consecutive integers is called an interval. If $\varphi$ is a proper edge $t$-coloring of a graph $G$ and $x\in V(G)$, then $S_G(x,\varphi)$ denotes the set of colors of edges of $G$ which are incident with $x$. A proper edge $t$-coloring $\varphi$ of a graph $G$ is called a cyclically-interval $t$-coloring if for any $x\in V(G)$ at least one of the following two conditions holds: a) $S_G(x,\varphi)$ is an interval, b) ${1,2,\ldots,t}\setminus S_G(x,\varphi)$ is an interval. For any $t\in \mathbb{N}$, let $\mathfrak{M}t$ be the set of graphs for which there exists a cyclically-interval $t$-coloring, and let $$\mathfrak{M}\equiv\bigcup{t\geq1}\mathfrak{M}_t.$$ For an arbitrary tree $G$, it is proved that $G\in\mathfrak{M}$ and all possible values of $t$ are found for which $G\in\mathfrak{M}_t.$

💡 Research Summary

The paper introduces a new edge‑coloring concept called cyclically‑interval edge‑coloring. For a proper edge‑coloring (\varphi : E(G)\rightarrow{1,\dots ,t}) of a simple connected graph (G), each vertex (x) has the set (S_G(x,\varphi)) of incident colors. The coloring is cyclically‑interval if for every vertex either (S_G(x,\varphi)) itself forms a consecutive integer interval, or its complement ({1,\dots ,t}\setminus S_G(x,\varphi)) does. In other words, the color palette is regarded as a circle, and at each vertex the used colors (or the unused colors) must occupy a single contiguous arc.

The authors focus on trees, the most elementary class of graphs, and answer two fundamental questions:

1. Existence – Does every tree admit a cyclically‑interval coloring for some number of colors?
2. Spectrum – For which integers (t) does a given tree admit such a coloring?

Let (\mathfrak{M}t) denote the family of graphs possessing a cyclically‑interval (t)-coloring, and (\mathfrak{M}=\bigcup{t\ge1}\mathfrak{M}_t). The main theorem proved is:

Theorem. For any tree (T) with maximum degree (\Delta(T)) and (|V(T)|) vertices, we have
\