Abstract
A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted bylak(G), is the least number of linear k-forests needed to decompose G. Recently, Zuo, He, and Xue studied the exact values of the linear(n−1)-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general k we show thatmax{lak(G),lal(H)}≤lamax{k,l}(G□H)≤lak(G)+lal(H)for any two graphs G and H. Denote byG∘H, G×HandG⊠Hthe lexicographic product, direct product and strong product of two graphs G and H, respectively. For any two graphs G and H, we also derive upper and lower bounds oflak(G∘H),lak(G×H)andlak(G⊠H)in this paper. The linear k-arboricity of a 2-dimensional grid graph, a r-dimensional mesh, a r-dimensional torus, a r-dimensional generalized hypercube and a hyper Petersen network are also studied.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
