Isolation toughness is a vital parameter to evaluate the vulnerability of computer networks. In specific network designing stage, it is necessary to find the lower bound of the isolated toughness, and strive to build a network that meets the stability requirements with the least cost. Gao et al.1 conjectured that if a graph G with κ ( G ) ≥ 3 m + 1 2 satisfies I ( G ) > 7 m + 5 4 m + 4 or I ′ ( G ) > 7 m + 5 4 m + 2 , then G is a ( P ≥ 3 , m ) -factor deleted graph. It's proved that this conjecture holds. However, it is found that as the connectivity changes, the tight lower bound of isolated toughness for ( P ≥ 3 , m ) -factor deleted graphs will change as well. Therefore, we propose a new perspective to look into this problem and introduce the concepts of isolated toughness ( P ≥ 3 , m ) factor deleted surface and isolated toughness variant ( P ≥ 3 , m ) factor deleted surface, where the result of the original conjecture is only a cross-section on surfaces. The main contribution in this paper is to determine the concrete expression of these two surfaces.

A 1-planar graph is a graph that can be drawn in the Euclidean plane such that each edge crosses at most one edge. An independent crossing (IC)-planar graph is a 1-planar graph satisfying the condition that two pairs of crossing edges have no common end-vertices. It is shown in this paper that every IC-planar graph is 6-choosable.

A plane graph $G$ is entirely $k$-colorable if $V(G)\cup E(G) \cup F(G)$ can be colored with $k$ colors such that any two adjacent or incident elements receive different colors. In 1993, Borodin proved that every plane graph $G$ with maximum degree $\Delta\ge 12$ is entirely $(\Delta+2)$-colorable. In this paper, we improve this result by showing that every plane graph $G$ with $\Delta\ge 9$ is entirely $(\Delta+2)$-colorable.