Liu Henry, Pikhurko Oleg, Sousa Teresa
CENTRO DE MATEMÁTICA E APLICAÇÕESFACULDADE DE CIÊNCIAS E TECNOLOGIA, UNIVERSIDADE NOVA DE LISBOACAMPUS DE CAPARICA2829-516CAPARICAPORTUGAL.
MATHEMATICS INSTITUTE AND DIMAPUNIVERSITY OF WARWICKCOVENTRY CV4 7ALUNITED KINGDOM.
J Graph Theory. 2015 Dec;80(4):287-298. doi: 10.1002/jgt.21851. Epub 2015 Jan 12.
Let be a graph whose edges are colored with colors, and H=(H1,⋯,Hk) be a -tuple of graphs. A H- of is a partition of the edge set of such that each part is either a single edge or forms a monochromatic copy of Hi in color , for some 1≤i≤k. Let φk(n,H) be the smallest number ϕ, such that, for every order- graph and every -edge-coloring, there is a monochromatic H-decomposition with at most ϕ elements. Extending the previous results of Liu and Sousa [Monochromatic Kr-decompositions of graphs, 76 (2014), 89-100], we solve this problem when each graph in H is a clique and n≥n0(H) is sufficiently large.
设(G)是一个其边用(k)种颜色染色的图，且(H=(H_1,\cdots,H_k))是一个(k)元组的图。(G)的一个(H)分解是(G)的边集的一个划分，使得每个部分要么是一条单边，要么在颜色(i)（对于某个(1\leq i\leq k)）中形成(H_i)的单色副本。设(\varphi_k(n,H))是最小的数(\varphi)，使得对于每个(n)阶图和每个(k)边染色，都存在一个至多有(\varphi)个元素的单色(H)分解。扩展刘和苏萨[图的单色(K_r)分解，《组合论杂志，B辑》76 (2014)，89 - 100]的先前结果，当(H)中的每个图都是完全图且(n\geq n_0(H))足够大时，我们解决了这个问题。
