Abstract
The Total Coloring Conjecture (TCC) states that every simple graph G is totally (Δ+2)-colorable, where Δ denotes the maximum degree of G. In this paper, we prove that TCC holds for dumbbell maximal planar graphs. Especially, we divide the dumbbell maximal planar graphs into three categories according to the maximum degree: J9, I-dumbbell maximal planar graphs and II-dumbbell maximal planar graphs. We give the necessary and sufficient condition for I-dumbbell maximal planar graphs, and prove that any I-dumbbell maximal planar graph is totally 8-colorable. Moreover, a linear time algorithm is proposed to compute a total (Δ+2)-coloring for any I-dumbbell maximal planar graph.
Highlights
IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
In [18], we study the total coloring of recursive maximal planar graphs and prove that Total Coloring Conjecture (TCC) is true for recursive maximal planar graphs
We aim to study the total coloring of dumbbell maximal planar graphs in this paper
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Regardless of the results in [13] or in [17], the graph G cannot contain adjacent triangles This leads us to study the total coloring and total chromatic number of maximal planar graphs, whose faces are all triangles. In [18], we study the total coloring of recursive maximal planar graphs and prove that TCC is true for recursive maximal planar graphs. (2,2)-recursive maximal planar graphs are totally (∆ + 1)-colorable. What is the total coloring of dumbbell maximal planar graphs? We aim to study the total coloring of dumbbell maximal planar graphs in this paper. In. Section 3, we prove that any dumbbell maximal planar graph is totally (∆ + 2)-colorable. In. Section 4, we propose an algorithm with linear time complexity to compute a total (∆ + 2)coloring for any I-dumbbell maximal planar graph.
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