Caterpillar arboricity of planar graphs

We solve a conjecture of Roditty, Shoham and Yuster [P.J. Cameron (Ed.), Problems from the 17th British Combinatorial Conference, Discrete Math., 231 (2001) 469-478; Y. Roditty, B. Shoham, R. Yuster, Monotone paths in edge-ordered sparse graphs, Discrete Math. 226 (2001) 411-417] on the caterpillar arboricity of planar graphs. We prove that for every planar graph G=(V,E), the edge set E can be partitioned into four subsets (E_i)_1?i?4 in such a way that G[E_i], for 1?i?4, is a forest of caterpillars. We also provide a linear-time algorithm which constructs for a given planar graph G, four forests of caterpillars covering the edges of G.     

Partitioning circulant graphs into isomorphic linear forests

1.IntroductionAlinearforestisaforestwhosecomponentsarepaths.Akiyama,E-coo,andHararyprovedthefollowingTheoremAll]andTheoremBIZ].TheoremA.Every3-regUlargraphGhasapartition(FI,F2)ofE(G)suchthatboth(V(G),FI)and(V(G),F2)arelinearforests.TheoremB.Every4reg...     

一类混合Ramsey数

设a(G)表示图G的点荫度,m为正整数,H为连通图,混合Ramsey数v(a;m;H)被定义的为最小的正整数P,使得对任意P阶图G则有a(G)≥m或者H包括于G^-。本文给出了v(a;m;H)的一种计算方法,并对图Cn和轮Wn确定了v(a;n;Cn)和v(a;m;Wn)的值。     

关于图的点荫度

本文研究了外平面图的点荫度和点荫度临界图,得到了外平面图G的点荫度a(G):a(G)={1,若G是树或林;2,若G不是树或林.     

Bounded degree acyclic decompositions of digraphs

An acyclic decomposition of a digraph is a partition of the edges into acyclic subgraphs. Trivially every digraph has an acyclic decomposition into two subgraphs. It is proved that for every integer s2 every digraph has an acyclic decomposition into s subgraphs such that in each subgraph the outdegree of each vertex v is at most . For all digraphs this degree bound is optimal.