Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$.  obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours necessary for such a colouring. One is a bound of $(2-\eps)\Delta^t$ for graphs of maximum degree at most $\Delta$, where $\eps$ is some absolute positive constant independent of $t$. The other is a bound of $O(\Delta^t/\log \Delta)$ (as $\Delta\to\infty$) for graphs of maximum degree at most $\Delta$ and girth at least $2t+1$. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least $g$, for every fixed $g \ge 3$, of arbitrarily large maximum degree $\Delta$, with distance-$t$ chromatic index at least $\Omega(\Delta^t/\log \Delta)$.
['Tomáš Kaiser', 'Ross J. Kang']
I am writing a research paper on a idea The distance-t chromatic index of graphs
[{"from": "human", "value": "I am writing a research paper on a idea The distance-t chromatic index of graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$.  obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours necessary for such a colouring. One is a bound of $(2-\\eps)\\Delta^t$ for graphs of maximum degree at most $\\Delta$, where $\\eps$ is some absolute positive constant independent of $t$. The other is a bound of $O(\\Delta^t/\\log \\Delta)$ (as $\\Delta\\to\\infty$) for graphs of maximum degree at most $\\Delta$ and girth at least $2t+1$. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least $g$, for every fixed $g \\ge 3$, of arbitrarily large maximum degree $\\Delta$, with distance-$t$ chromatic index at least $\\Omega(\\Delta^t/\\log \\Delta)$."}]