The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, t...The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with △（G）≥｜G｜ - 2≥9 has Xef（G） = △（G）.展开更多
Let f be a proper edge coloring of G using k colors. For each x ∈ V（G）, the set of the colors appearing on the edges incident with x is denoted by Sf（x） or simply S（x） if no confusion arise. If S（u） = S（v） ...Let f be a proper edge coloring of G using k colors. For each x ∈ V（G）, the set of the colors appearing on the edges incident with x is denoted by Sf（x） or simply S（x） if no confusion arise. If S（u） = S（v） and S（v） S（u） for any two adjacent vertices u and v, then f is called a Smarandachely adjacent vertex distinguishing proper edge col- oring using k colors, or k-SA-edge coloring. The minimum number k for which G has a Smarandachely adjacent-vertex-distinguishing proper edge coloring using k colors is called the Smarandachely adjacent-vertex-distinguishing proper edge chromatic number, or SA- edge chromatic number for short, and denoted by Xsa（G）. In this paper, we have discussed the SA-edge chromatic number of K4 V Kn.展开更多
In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the doub...In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the double inductions and the method of exchanging colors from the aspect of configuration property. For series-parallel graphs of △(G) ≥ 5, △(G) ≤ x'as(G) ≤ △(G) + 1. Moreover, x'as(G) = △(G) + 1 if and only if it has two adjacent vertices of maximum degree, where △(G) and X'as(G) denote the maximum degree and the adjacent strong edge chromatic number of graph G respectively.展开更多
A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges ...A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to v, where uv ∈E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ＇αα（G）, is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G（V, E） is a graph with no isolated edges, then χ＇αα（G）≤32△.展开更多
ALL graphs appearing in this note are simple. A graph with p vertices and q edges will becalled a (p, q )-graph. The maximum degree of G is denoted by Δ(G).Let n≥2 be an integer. The n-edge chromatic number X_n (G) ...ALL graphs appearing in this note are simple. A graph with p vertices and q edges will becalled a (p, q )-graph. The maximum degree of G is denoted by Δ(G).Let n≥2 be an integer. The n-edge chromatic number X_n (G) of a simple graph G is theminimum cardinality of a set of colors with which one can assign the colors to the edges of Gsuch that the edges on a path of length less than or equal to n receive different colors.The aim of this note is to explore the bounds for X’_n (G) and X’_n (G) + X’_n (G). It is展开更多
基金This research is supported by NNSF of China(40301037, 10471131)
文摘The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with △（G）≥｜G｜ - 2≥9 has Xef（G） = △（G）.
基金Supported by NNSF of China(61163037,61163054,61363060)
文摘Let f be a proper edge coloring of G using k colors. For each x ∈ V（G）, the set of the colors appearing on the edges incident with x is denoted by Sf（x） or simply S（x） if no confusion arise. If S（u） = S（v） and S（v） S（u） for any two adjacent vertices u and v, then f is called a Smarandachely adjacent vertex distinguishing proper edge col- oring using k colors, or k-SA-edge coloring. The minimum number k for which G has a Smarandachely adjacent-vertex-distinguishing proper edge coloring using k colors is called the Smarandachely adjacent-vertex-distinguishing proper edge chromatic number, or SA- edge chromatic number for short, and denoted by Xsa（G）. In this paper, we have discussed the SA-edge chromatic number of K4 V Kn.
基金National Natural Science Foundation of China (60103021, 60274026)
文摘In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the double inductions and the method of exchanging colors from the aspect of configuration property. For series-parallel graphs of △(G) ≥ 5, △(G) ≤ x'as(G) ≤ △(G) + 1. Moreover, x'as(G) = △(G) + 1 if and only if it has two adjacent vertices of maximum degree, where △(G) and X'as(G) denote the maximum degree and the adjacent strong edge chromatic number of graph G respectively.
基金Supported by the Natural Science Foundation of Gansu Province(3ZS051-A25-025)
文摘A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to v, where uv ∈E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ＇αα（G）, is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G（V, E） is a graph with no isolated edges, then χ＇αα（G）≤32△.
文摘ALL graphs appearing in this note are simple. A graph with p vertices and q edges will becalled a (p, q )-graph. The maximum degree of G is denoted by Δ(G).Let n≥2 be an integer. The n-edge chromatic number X_n (G) of a simple graph G is theminimum cardinality of a set of colors with which one can assign the colors to the edges of Gsuch that the edges on a path of length less than or equal to n receive different colors.The aim of this note is to explore the bounds for X’_n (G) and X’_n (G) + X’_n (G). It is
