A graph is 1-toroidal, if it can be embedded in the torus so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-toroidal graph with maximum degree Δ≥ 10 is of class one in terms of edge coloring. Meanwhile, we show that there exist class two 1-toroidal graphs with maximum degree Δ for each Δ≤ 8.
In this paper,we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group(Δ(G)+1)-edge-choosable,and some planar graphs with large girth and maximum degree are groupΔ(G)-edge-choosable.
A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2∨(K1∪K2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1∨(K1∪K2 ) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.
A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has(1)lc(G) ≤Δ 21 if Δ≥ 9; (2)lc(G) ≤「Δ/2」 + 7 ifg ≥ 5; (3) lc(G) ≤「Δ/2」 + 2 ifg ≥ 7 and Δ≥ 7.
让 G 是连接 matroid 的一张电路图。P. 李和 G. 刘[Comput。数学。Appl， 2008， 55：654659 ] 证明 G 让哈密尔顿包括 e 和如果 G 有至少四个顶点，为 G 的任何边 e 排除 e 的另一个哈密尔顿周期骑车。这份报纸证明 G 有一个哈密尔顿周期包括 e 并且如果 G 有至少五个顶点，为任何二边 e 和 G 的 e 排除 e。这结果是在某感觉可能的最好。一个开的问题这份报纸最后被建议。
