A Note on Decycling Number, Vertex Partition and AVD-Total Coloring in Graphs

Abstract

This note provides a new perspective, i.e., graph embeddings on the decycling number $\nabla (G)$ (Beineke and Vandell in J Graph Theory 25:59–77, 1997) of a graph G. For this point, it is shown that $\nabla (G)=\gamma _M(G)+\xi (G)$ for any cubic graph G and $|S|=\frac{\beta (G)+m(S)}{k-1}$ for any decycling set S of a k-regular graph G, where $\gamma _M(G)$, $\xi (G)$, $\beta (G)$ and $m(S)=c(G-S)+|E(S)|-1$ ($c(G-S)$ is the number of components of $G-S$ and |E(S)| is the number of edges in a subgraph G[S]) are, respectively, the maximum genus, the Betti deficiency (Xuong in J Combin Theory Ser B 26:217–225, 1979), the cycle rank (Harary in Graph theory, Academic Press, New York, 1967) and the margin number of G. Meanwhile, we further confirm that (1) a cubic graph G ($G\ne K_4$) has a vertex partition $(V_1, V_2)$ such that $V_1$ is an independent set and $V_2$ induces a forest and (2) a k-regular graph G with $\nabla (G)=\frac{\beta (G)+m(S)}{k-1}$ ($m(S)\le 2$) has a vertex partition $(S,G-S)$ such that G[S] contains at most two edges and $G-S$ induces a forest, where S is the smallest decycling set of G. Resorting to the above vertex partitions, we get the adjacent vertex distinguishing (AVD) total chromatic numbers of some families of graphs, and these results verify Zhang’s conjecture (Zhang in Sci China Ser A 48:289–299, 2005) that every graph with maximum degree $\Delta$ has an AVD-total $(\Delta +3)$-coloring.