Two-distance and list-two-distance coloring of cacti: the subcubic case and the C5 obstruction

RESUMO

The square $G^2$ of a graph joins two vertices at distance at most two; a proper coloring of $G^2$ is a 2-distance coloring of $G$. For a cactus $G$ (every edge on at most one cycle) we determine both the 2-distance chromatic number $\chi(G^2)$ and the choice number $\mathrm{ch}(G^2)$ exactly: they are always equal, and the common value is $\Delta+1$ if $\Delta\ge4$, is 4 if $\Delta=3$ and $G$ has no block equal to $C_5$, is 5 if $\Delta=3$ and $G$ has a $C_5$ block, and is the classical value if $\Delta\le2$. For $\Delta\ge6$ the value $\Delta+1$ is already known, since cacti are outerplanar and hence $K_{2,3}$-minor-free (Hetherington-Woodall; Agnarsson-Halldorsson). Our contribution is the small-degree regime. For subcubic cacti we obtain a complete classification in both the ordinary and list settings, with the 5-cycle as the unique obstruction; the list statement has no prior analogue and is a genuine positive instance of the List Square Coloring Conjecture, which is false in general. A single elimination order then handles all $\Delta\ge4$ uniformly and, in particular, settles the two cases $\Delta\in\{4,5\}$ that the superclass bounds leave at $\Delta+2$. The number 5 turns out to be one obstruction wearing three disguises: $C_5^2=K_5$, the Frobenius number of $\{3,4\}$, and a degenerate $K_4$ list-coloring instance.