It has been shown that there is a planar graph without 3-cycles which is not $({\Delta }_1,{\Delta }_2)$-colorable for any given ${\Delta }_1,{\Delta }_2$. This inspires many research to obtain sufficient conditions for planar graphs without 4-cycles and other cycles to be $({\Delta }_1,{\Delta }_2)$-colorable. For example, planar graphs without k-cycles and 4-cycles are $({\Delta }_1,{\Delta }_2)$-colorable for each $k\in \{3,5\}$ and $({\Delta }_1,{\Delta }_2)=\{(2,6),(3,4)\}$. In this work, we study the values of ${\Delta }_1$ and ${\Delta }_2$ that make a planar graph without only 4-cycles $({\Delta }_1,{\Delta }_2)$-colorable. We begin with ${\Delta }_1=6$ and ${\Delta }_2=6$. Planar graphs without 4-cycles are $(6,6)$-colorable. Two colors in a $({\Delta }_1,{\Delta }_2)$-coloring, where ${\Delta }_1={\Delta }_2$ are switchable, thus reflect the symmetry of the resulting coloring. Furthermore, some proof techniques are using the symmetry of these two colors.

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.

An independent dominating set I of a graph G is an independent set such that every vertex in $V\left(G\right)-I$ has at least one neighbor in I. The independent domination number of G is the minimum cardinality of an independent dominating set of G. The independent bondage number of a graph G, denoted by $b_i\left(G\right)$, is the minimum cardinality of an edge set whose removal increases the independent domination number of G. In 2025, Gamlath et al. proved that for planar graphs G with $\delta \left(G\right)\ge 2$, the following holds: $b_i\left(G\right)\le 5$ if G does not contain cycles of lengths 3 and 4, and $b_i\left(G\right)\le 4$ if G does not contain cycles of lengths 3, 4, 5, and 6. This motivates us to investigate the independent bondage number of planar graphs with $\delta \left(G\right)\ge 2$ without cycles of lengths 4 and 5, or cycles of lengths 4, 5, 6, and 7. In this paper, we prove the following bounds for planar graphs G with $\delta \left(G\right)\ge 2$: $b_i\left(G\right)\le 5$ if G does not contain cycles of lengths 4 and 5, and $b_i\left(G\right)\le 4$ if G does not contain cycles of lengths 4, 5, 6, and 7.

A defectivek-coloring is a coloring on the vertices of a graph with colors $1,2,\dots ,k$ where adjacent vertices may have the same color. A $(d_1,d_2)$-coloring of a graph G is a defective two-coloring such that each vertex colored by i has at most $d_i$ adjacent vertices of the same color, where $i=1,2$. A graph G is $(d_1,d_2)$-colorable if it admits $(d_1,d_2)$-coloring. For planar graphs excluding cycles with three, four, and six vertices, Dross and Ochem, and additionally Sittitrai and Pimpasalee, have studied their defective 2-coloring. They showed that such graphs are $(0,6)$- and $(3,3)$-colorable, respectively. We show in this work that these graphs are also $(2,4)$-colorable.