Zero Forcing and Vertex Independence Number on Cubic and Subcubic Graphs

Abstract

Motivated by a conjecture from the automated conjecturing program TxGraffiti, in this paper the relationship between the zero forcing number, $Z(G)$, and the vertex independence number, $α(G)$, of cubic and subcubic graphs is explored. TxGraffiti conjectures that for all connected cubic graphs $G$, that are not $K_4$, $Z(G) \leq α(G) + 1$. This work uses decycling partitions of upper-embeddable graphs to show that almost all cubic graphs satisfy $Z(G) \leq α(G) + 2$, provides an infinite family of cubic graphs where $Z(G) = α(G) + 1$, and extends known bounds to subcubic graphs.

Figures (5)

• Figure 1: A longest path $c_0a_0c_1a_1\dots c_ka_k$ that alternates between an independent set $A$ and components in $G-A$.
• Figure 2: A leaf of $T$ replaced by a $K_4$ with a subdivided edge.
• Figure 3: Graphs $G$ and $G_1(v)$.
• Figure 4: Graphs $G$ and $G_2(v)$.
• Figure 5: Graphs $G$ and $G'(v)$.

Theorems & Definitions (42)

• Conjecture 1: BDSY
• Theorem 2: DH
• Lemma 3
• proof
• Theorem 4: konig1916
• Theorem 5: pathcover
• Theorem 6
• proof
• Corollary 7
• proof