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Rainbow Monochromatic k -Edge-Connection Colorings of Graphs
Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2021-03-27 , DOI: 10.1007/s00373-021-02304-x
Ping Li , Xueliang Li

A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call k paths \(P_1,\ldots ,P_k\) rainbow monochromatic paths if every \(P_i\) is monochromatic and for any two \(i\ne j\), \(P_i\) and \(P_j\) have different colors. An edge-coloring of a graph G is said to be a rainbow monochromatic k-edge-connection coloring (or \(RMC_k\)-coloring for short) if every two distinct vertices of G are connected by at least k rainbow monochromatic paths. We use \(rmc_k(G)\) to denote the maximum number of colors that ensures G has an \(RMC_k\)-coloring, and this number is called the rainbow monochromatic k-edge-connection number. We prove the existence of \(RMC_k\)-colorings of graphs, and then give some bounds of \(rmc_k(G)\) and present some graphs whose \(rmc_k(G)\) reaches the lower bound. We also obtain the threshold function for \(rmc_k(G(n,p))\ge f(n)\), where \(\left\lfloor \frac{n}{2}\right\rfloor > k\ge 1\).



中文翻译:

图的彩虹单色k-边缘连接着色

如果边缘彩色图中的所有路径具有相同的颜色,则该路径称为单色路径。如果每个\(P_i \)是单色的并且对于任何两个\(i \ ne j \)\(P_i \)\(P_j \)我们将k条路径称为(P_1,\ ldots,P_k \)彩虹单色路径有不同的颜色。边缘着色图的ģ被说成是一个彩虹单色ķ _edge时连接的着色(或\(RMC_k \) -coloring的简称)如果每两个不同的顶点ģ通过至少连接ķ彩虹单色路径。我们使用\(rmc_k(G)\)表示确保G具有\(RMC_k \)着色的最大颜色数,该数字称为彩虹单色k边缘连接数。我们证明了\(RMC_k \) -图的着色的存在,然后给出了\(rmc_k(G)\)的一些界限,并给出了\(rmc_k(G)\)达到下界的一些图。我们还获得\(rmc_k(G(n,p))\ ge f(n)\)的阈值函数,其中\(\ left \ lfloor \ frac {n} {2} \ right \ rfloor> k \ ge 1 \)

更新日期:2021-03-27
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