A Structure of 1-Planar Graph and Its Applications to Coloring Problems
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A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the (p, 1)-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the (p, 1)-total labelling number of every 1-planar graph G is at most \(\Delta (G)+2p-2\) provided that \(\Delta (G)\ge 8p+2\) and \(p\ge 2\), and show that every 1-planar graph has an equitable edge coloring with k colors for any integer \(k\ge 18\). These three results respectively generalize the main theorems of three different previously published papers.
Keywords1-Planar graph List edge coloring List total coloring (p, 1)-Total labelling Equitable edge coloring
Mathematics Subject Classification05C15 05C10
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