The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove tha...The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords.展开更多
A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with...A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with Δ（G） ≥ 7 and without chordal 7-cycles,then G has a（Δ（G） ＋ 1）-total-coloring.展开更多
The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum deg...The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9.展开更多
文摘The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords.
基金Supported by National Natural Science Foundation of China(Grant No.11271006)
文摘A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with Δ（G） ≥ 7 and without chordal 7-cycles,then G has a（Δ（G） ＋ 1）-total-coloring.
基金supported by Natural Science Foundation of Shandong Province (Grant No. ZR2009AM009)Scientific Research Foundation for the Excellent Middle-Aged and Youth Scientists of Shandong Province (Grant No. BS2012SF016)National Natural Science Foundation of China (Grant Nos.11001055 and 11101243)
文摘The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9.
