A mixed graph has a set of vertices, a set of undirected egdes, and a set of directed arcs. A proper coloring of a mixed graph $G$ is a function $c$ that assigns to each vertex in $G$ a positive integer such that, for each edge $uv$ in $G$, $c(u) \ne c(v)$ and, for each arc $uv$ in $G$, $c(u) < c(v)$. For a mixed graph $G$, the chromatic number $\chi(G)$ is the smallest number of colors in any proper coloring of $G$. A directional interval graph is a mixed graph whose vertices correspond to intervals on the real line. Such a graph has an edge between every two intervals where one is contained in the other and an arc between every two overlapping intervals, directed towards the interval that starts and ends to the right. Coloring such graphs has applications in routing edges in layered orthogonal graph drawing according to the Sugiyama framework; the colors correspond to the tracks for routing the edges. We show how to recognize directional interval graphs, and how to compute their chromatic number efficiently. On the other hand, for mixed interval graphs, i.e., graphs where two intersecting intervals can be connected by an edge or by an arc in either direction arbitrarily, we prove that computing the chromatic number is NP-hard.
翻译：混合图有一套顶点、 一组未方向的顶点和一组定向弧。 混合图$G$的正确颜色是一个函数 $c$, 给每个顶点指定一个正整数, 以美元表示每个边缘$uv$$$$$$, 美元( u)\ ne c( v) 美元, 以美元计, 美元( u) < c( v) 美元。 对于混合图$ G$, 色数$\ chi( G) $( $) 是任何适当颜色中最小的颜色数 $G$。 方向图是一个混合图, 其顶点与实际线上的间距相符 $G$$, 美元( u) c( un) c( v) 美元), 而对于每两个折点的折点之间有一个边缘, 以美元( $G$( $) $( u) $( u) < c( c)( c) < c( v) ) ) ) river( c( c( viver) river) river) river( g) rut) $) $( ) $( $) $) $( $) $( $) $) ) $( $( g) $) $( g) $( g) $) $( $( $( g) $) $( $) $( g) $( g) $( g) ) $) ) ) $( $( $( g) ) ) $( $( ) ) $( $( $) ) $( $) ) ) $( ) $( $) $) $) $) $( $( $( $) $) $) $( $) $( $) $) ) $( ) $) ) $( $( ) $( $( $) $) $) $( $) $
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