An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval $t$-coloring} if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. In 1990, Kamalian proved that if a graph $G$ with at least one edge has an interval $t$-coloring, then $t\leq 2|V(G)|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{11}{6}|V(G)|$. In the same paper Axenovich suggested a conjecture that if a planar graph $G$ has an interval $t$-coloring, then $t\leq \frac{3}{2}|V(G)|$. In this paper we confirm the conjecture by showing that if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{3|V(G)|-4}{2}$. We also prove that if an outerplanar graph $G$ has an interval $t$-coloring, then $t\leq |V(G)|-1$. Moreover, all these upper bounds are sharp.

翻译：图 $G$ 的边着色使用颜色 $1,\ldots,t$，若所有颜色均被使用，且与 $G$ 中每个顶点关联的边颜色互异并构成整数的区间，则称为 *区间 $t$-着色*。1990 年，Kamalian 证明：若至少有一条边的图 $G$ 存在区间 $t$-着色，则 $t\leq 2|V(G)|-3$。2002 年，Axenovich 针对平面图改进了这一上界：若平面图 $G$ 允许区间 $t$-着色，则 $t\leq \frac{11}{6}|V(G)|$。在同一论文中，Axenovich 提出猜想：若平面图 $G$ 存在区间 $t$-着色，则 $t\leq \frac{3}{2}|V(G)|$。本文通过证明若平面图 $G$ 允许区间 $t$-着色，则 $t\leq \frac{3|V(G)|-4}{2}$ 来确认该猜想。我们还证明若外平面图 $G$ 存在区间 $t$-着色，则 $t\leq |V(G)|-1$。此外，所有这些上界都是紧的。