How efficiently can a closed curve of unit length in \(\mathbb{R}^d\) be covered by \(k\) closed curves so as to minimize the maximum length of the \(k\) curves? We show that the maximum length is at most \(2k^{-1} – \frac{1}{4} k^{-4}\) for all \(k\geq 2\) and \(d \geq 2\). As a first byproduct, we show that \(k\) agents can traverse a Euclidean TSP instance significantly faster than a single agent. We thereby sharpen recent planar results by Berendsohn, Kim, and Kozma (2025) and extend these improvements to all dimensions. As a second byproduct, we obtain a linear time approximation algorithm with ratio \(2 – \frac{1}{4} k^{-3}\) for covering any closed polygonal curve in \(\mathbb{R}^d\) by \(k\) closed curves so that the maximum length of an individual curve is minimized.

A Fibonacci cordial (FC) labeling of a graph G is an injective function f : V(G) → {F₀, F₁, …, F_n}, where F_i is the i^th Fibonacci number, such that the induced edge labeling f^*(uv) = (f(u) + f(v)) (mod  2) satisfies |e_f(0) − e_f(1)| ≤ 1. A graph admitting such a labeling is called a Fibonacci cordial. First introduced by Rokad and Ghodasara (2016), FC labeling has been studied for several graph families. Mitra and Bhoumik (2020) extended this to complete graphs, cycles, and their corona (C_n and K_p for p ≤ 3). Motivated to build upon their work, we investigate C_n ⊙ K_p for p ≥ 4. Additionally, we examine whether the aforementioned family of corona graphs retains Fibonacci cordiality when alterations are made to the order of the corona, as observed in the family K_n ⊙ C_m. Moreover, we investigate the conditions under which two additional corona graph families, namely K_m ⊙ K_m and K_n, n ⊙ K_p, exhibit Fibonacci cordial labeling.

A graph is near-bipartite if its vertex set can be partitioned into two parts such that one part is an independent set and the other induces a forest. It is clear that near-bipartite graphs are 3-colorable. It was proved that planar graphs without 4-, 5– and 8-cycles are 3-colorable [Discuss. Math. Graph Theory, 31(2011): 775-789]. It is asked by Kang et al that whether it is true that the planar graphs without 4-, 5– and 8-cycles are near-bipartite [Discuss. Math. Graph Theory 45 (2025) 129-145]. A 6-cycle is called a special 6-cycle if this 6-cycle shares an edge with a triangle of G. In this paper, we prove that planar graphs without 4-, 5-, 8-cycles and special 6-cycles are near-bipartite which is a step toward the problem.

We study a modular generalization of the (σ, ρ)-Dominating Set problem on graphs of bounded treewidth, where vertices must satisfy neighborhood constraints modulo a fixed integer m. We present a dynamic programming algorithm over tree decompositions that achieves a runtime of O(n ⋅ (2m)^2tw + 2) and space usage O(n ⋅ (2m)^tw + 1), establishing fixed-parameter tractability when both treewidth tw and log m are treated as parameters. We prove this bound exhibits the correct exponential dependence on twlog m, as this factor is inherent to modular constraint satisfaction under the Strong Exponential Time Hypothesis (SETH). Experimental evaluation on synthetic graphs confirms the algorithm’s efficiency for small values of tw and m, highlighting its applicability to network design, logic circuits, and distributed systems with modular constraints.

The solutions of some new triangular designs not tabulated in Clatworthy (1973) are presented here. These designs are known in the literature but re–presented here with more explicit block lists. As a by–product, resolvable solutions of some designs are also obtained. The work addresses a recognized gap in combinatorial design theory and appears to extend classical catalogs.

Let W = {w₁, w₂, w₃, …, w_k} be an ordered set of vertices in a connected graph G. The representation of a vertex v ∈ V(G) with respect to W is the k-tuple r(v|W) = (d(v, w₁), d(v, w₂), …, d(v, w_k)), where d(v, w_i) is the length of the shortest path from v to w_i. If each vertex in G is uniquely identified by the distance vector, r(v|W) = (d(v, w₁), d(v, w₂),…,d(v, w_k)), then W is called a resolving set for G. If the resolving set is also independent, it is referred to as an independent resolving set. The independent metric dimension of G, denoted by idim(G), is the smallest cardinality of an independent resolving set. This study explores the independent metric dimension of the circulant graphs C_n(1, 2), C_n(1, 2, 3), C_n(1, 2, 3, 4) for sufficiently large n.

In this paper, given a homeomorphism f of a compact metric space X, we show that the set of all asymptotic average shadowable points of f is an open and invariant set and f has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of f is X if and only if any Borel probability measure μ of X has the asymptotic average shadowing property.

A supermagic labeling (often also called vertex-magic edge labeling) of a graph \(G(V,E)\) with \(|E|=q\) is a bijection from \(E\) to the set of first \(k\) positive integers such that the sum of labels of all incident edges of every vertex \(x\in V\) is equal to the same integer \(c\). An existence of a supermagic labeling of Cartesian product of two cycles, \(C_{n}\Box C_m\) for \(n,m\geq4\) and both \(n,m\) even and for any \(C_n\Box C_n\) with \(n\geq3\) was proved by Ivančo. Ivančo also conjectured that such labeling is possible for any \(C_n\Box C_m\) with \(n,m\geq3\). We prove his conjecture for all \(n,m\) odd that are not relatively prime.

Let \(G=(V,E)\) be a graph. For a vertex \(u\) of \(V(G)\), its closed neighborhood, \(N[S]\), is defined as \(N[u]=\{u\}\cup\{v|v\in V(G), v\neq u, u\) and \(v\) are adjacent in \(G \}\). A vertex subset \(S\) of \(V(G)\) is called a subversion strategy of \(G\) if each of the vertices in \(N[S]\) is deleted from \(G\). By \(G/S\) we denote the survival subgraph \(G-N[S]\). A subversion strategy \(S\) is called a cut strategy of \(G\) if \(G/S\) is disconnected, or is a clique, or is empty. In this paper, we revise the definition of neighbor-isolated scattering number, which was introduced by Aslan, as \(NIS(G)=\max\{i(G/S)-|S|\}\), where \(S\) represents a cut strategy of \(G\) such that every component of \(G/S\) is an isolated vertex or a clique, and \(i(G/S)\) represents the number of the components of \(G/S\). We discuss the relationship between this parameter and the structure of graphs. Some tight bounds and extremal graphs with given order and neighbor-isolated scattering number are determined.

Let \(k\) be an odd prime and choose \(s\in\mathbb{Z}_k^\times\) with \(s^2\not\equiv \pm1\pmod{k}\) (hence \(k\ge7\)). We give a deterministic, purely algebraic construction of compound pandiagonal (Nasik) magic squares of order \(k^{2}\) with consecutive entries \(\{0,1,\dots,k^{4}-1\}\). The input is the \(k\times k\) Modular Inverse Shift (MIS) kernel \(M_s(i,j)=si+s^{-1}j\in\mathbb{Z}_k\), a classical linear Latin square. Our contribution is not a new Latin-square object, but a closed-form integration of: (i) orthogonality of \((M_s,M_s^{\mathsf T})\), (ii) toroidal diagonal-regularity, and (iii) a two-level base-\(k\) digit superposition producing a \(k^2\times k^2\) square with closed-form evaluation of entries. We encode four \(\mathbb{Z}_k\)-digits coming from \((M_s,M_s^{\mathsf T})\) at both the block level and the within-block level, obtaining an explicit formula \(P_s(I,J)\in\{0,\dots,k^{4}-1\}\). Orthogonality yields bijectivity, while a carry-sensitive diagonal decomposition proves that every broken diagonal of both slopes sums to the magic constant. Finally, evaluating block sums shows that the induced \(k\times k\) block-sum array is itself pandiagonal magic, establishing the compound property.