Given a graph G, a function c : V(G) →{1,..., k} with the property that for every u≠v, c(u) = c(v) = i implies that the distance between u and v is greater than i, is called a k-packing coloring of G. The smallest integer k for which there exists a k-packing coloring of G is called the packing chromatic number of G, and is denoted by χρ(G). Packing chromatic vertex-critical graphs are the graphs G for which χρ(G − x) < χρ(G) holds for every vertex x of G. A graph G is called a packing chromatic critical graph if for every proper subgraph H of G, χρ(H) < χρ(G). Both of the mentioned variations of critical graphs with respect to the packing chromatic number have already been studied [6, 23]. All packing chromatic (vertex-)critical graphs G with χρ(G) = 3 were characterized, while there were known only partial results for graphs G with χρ(G) = 4. In this paper, we provide characterizations of all packing chromatic vertex-critical graphs G with χρ(G) = 4 and all packing chromatic critical graphs G with χρ(G) = 4.

Let G be a graph and let S = (s1,s2,..., sk) be a non-decreasing sequence of positive integers. An S-packing coloring of G is a mapping c : V(G) → {1, 2,..., k} with the following property: if c(u) = c(v) = i, then d(u,v) > si for any i ∈ {1, 2,...,k}. In particular, if S = (1, 2, 3, ..., k), then S-packing coloring of G is well known under the name packing coloring. Next, let r be a positive integer and u,v ∈ V(G). A vertex u r-distance dominates a vertex v if dG(u, v)≤ r. In this paper, we present a new concept of a coloring, namely distance dominator packing coloring, defined as follows. A coloring c is a distance dominator packing coloring of G if it is a packing coloring of G and for each x ∈ V(G) there exists i ∈ {1,2, 3,...} such that x i-distance dominates each vertex from the color class of color i. The smallest integer k such that there exists a distance dominator packing coloring of G using k colors, is the distance dominator packing chromatic number of G, denoted by χdρ(G). In this paper, we provide some lower and upper bounds on the distance dominator packing chromatic number, characterize graphs G with χdρ(G) ∈ {2,3}, and provide the exact values of χdρ(G) when G is a complete graph, a star, a wheel, a cycle or a path. In addition, we consider the relation between χρ (G) and χdρ(G) for a graph G.

Given a graph G and a positive integer i, an i-packing in G is a subset W of the vertex set of G such that the distance between any two distinct vertices from W is greater than i. The packing chromatic number of a graph G, χρ(G), is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ {1,..., k}, where each Vi is an i-packing. In this paper, we present some general properties of packing chromatic numbers of finite super subdivisions of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and some neighborhood corona graphs.

Whole number addition and subtraction are required topics in early-grade mathematics curricula. This study, with a sample size of N=117, introduces a unique method of using self-designed drawings to explore visual representations of mathematical concepts. The collected drawings show how preservice teachers communicate mathematical ideas, and reveal insights into their grasp of the place value concept and potential addition/subtraction teaching visual representations. Results showed that even though a significant portion of the participants demonstrated a clear comprehension of how to teach key mathematical concepts, a prominent trend emerged: about one-third of preservice teachers did not employ base ten grouping in their drawings of two-digit numbers. Similarly, one-third illustrated arithmetic operations symbolically, merely converting numbers into iconic forms, and additionally often misrepresented subtraction. These findings point to specific areas where preservice teachers’ meta-representational abilities could be improved. By emphasizing and strengthening these areas in teacher education programs, the pedagogical skills of future educators can be enhanced.