Total Forcing and Zero Forcing in Claw-Free Cubic Graphs

TL;DR

This paper investigates the total forcing number in specific classes of cubic graphs, establishing upper bounds and characterizing extremal graphs, with implications for dynamic coloring processes.

Contribution

It proves upper bounds for the total forcing number in connected claw-free cubic graphs and characterizes the graphs where equality holds.

Findings

For graphs with a spanning 2-factor of triangles, the total forcing number is at most half the number of vertices.

In connected, claw-free, cubic graphs of order at least 6, the total forcing number is at most half the number of vertices.

The extremal graphs achieving these bounds are explicitly characterized.

Abstract

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set (zero forcing set) of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The total forcing number of $G$, denoted $F_t(G)$, is the minimum cardinality of a total forcing set in $G$. We prove that if $G$ is a connected cubic graph of order~$n$ that has a spanning $2$-factor consisting of triangles, then $F_t(G) \le \frac{1}{2}n$. More generally, we prove that if $G$ is a connected, claw-free, cubic graph of order~$n \ge 6$, then $F_t(G) \le \frac{1}{2}n$, where a claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph. The graphs achieving equality in these bounds are characterized.