The circular chromatic index of graphs of high girth (Q858677)

A proper circular \(l\)-coloring of a graph \(G\), for real \(l\geq1\), is a coloring of the vertices of \(G\) by real numbers from the interval \([0,l)\), such that the difference modulo \(l\) of the colors \(\gamma_1\) and \(\gamma_2\) assigned to two adjacent vertices is at least one, i.e. \(1\leq| \gamma_1-\gamma_2| \leq l-1\). The smallest real \(l\) for which there is a proper circular \(l\)-coloring is called the circular chromatic number of \(G\). The circular chromatic index \(\chi^{'}_c(G)\) of \(G\) is defined to be the circular chromatic number of \(L(G)\), the line graph of the graph \(G\). The main result of the paper is the following one: For any integer \(\Delta\geq1\) and real \(\varepsilon>0\), there exists a positive integer \(g\) such that if \(G\) is a graph of maximum degree \(\Delta\) and girth at least \(g\), then \( \chi^{'}_c(G)\leq\Delta+\varepsilon\,. \)