$(2,4)$-Colorability of Planar Graphs Excluding $3$-, $4$-, and $6$-Cycles
Pongpat Sittitrai, Wannapol Pimpasalee, Kittikorn Nakprasit
TL;DR
The paper proves that planar graphs with no $3$-, $4$-, or $6$-cycles admit a $(2,4)$-coloring, i.e., a defective $2$-coloring where color $1$ vertices have at most two same-color neighbors and color $2$ vertices at most four. Building on prior defective-coloring results for such graphs, it uses a minimal-counterexample and a discharging method with carefully designed vertex/face rules to rule out all configurations, culminating in a contradiction, thereby establishing the result. The work strengthens the understanding of defective colorability under cycle-avoidance constraints and contributes toward the conjecture that these graphs are $(d_1,d_2)$-colorable whenever $d_1+d_2\ge6$. The approach illustrates how structural lemmas and charge redistribution can certify colorability properties in planar graphs with restricted cycle structures.
Abstract
A defective $k$-coloring is a coloring on the vertices of a graph using colors $1,2, \dots, k$ such that adjacent vertices may share the same color. A $(d_1,d_2)$-\emph{coloring} of a graph $G$ is a defective $2$-coloring of $G$ such that any vertex colored by color $i$ has at most $d_i$ adjacent vertices of the same color, where $i\in\{1,2\}$. A graph $G$ is said to be $(d_1,d_2)$-\emph{colorable} if it admits a $(d_1,d_2)$-coloring. Defective $2$-coloring in planar graphs without $3$-cycles, $4$-cycles, and $6$-cycles has been investigated by Dross and Ochem, as well as Sittitrai and Pimpasalee. They showed that such graphs are $(0,6)$-colorable and $(3,3)$-colorable, respectively. In this paper, we proved that these graphs are also $(2,4)$-colorable.