Every Cubic Bipartite Graph has a Prime Labeling Except $K_{3,3}$

Abstract

A graph G is prime if the vertices can be distinctly labeled with the integers $1,2, \ldots ,|V(G)|$ so that adjacent vertices have relatively prime labels. We show that every cubic bipartite graph is prime except $K_{3,3}$, which implies a number of other results. We also provide evidence to support a conjectured classification for the primality of 2-regular graphs.

Appendix A: The Tables

See Tables 1, 2, 3 and 4.

Table 1 Values confirming the inequality $7+A_\alpha -3B_\alpha -\left\lfloor \alpha \right\rfloor +C_\alpha \ge 0$ from Lemma 3 for $\alpha =i/2$, $i=3,4, \ldots ,34$

Table 2 Calculations showing the maximum value $x=\left\lfloor 2N/p\right\rfloor$ needed to test in the proof of Corollary 1

Table 3 Required values of f(n) for use in the proof of Corollary 1, where $n=\lceil xp/2\rceil$ and $x\le 13$

Table 4 Required values of f(n) for use in the proof of Corollary 1, where $n=\lceil xp/2\rceil$ and $x\ge 14$