A graph \(G=(V,E)\) is said to be an absolute mean graceful graph if there exists a one-to-one function \(f:V(G)\to \{0,\pm1,\pm2,\ldots,\pm|E(G)|\}\) such that the induced edge-labeling function \(f^*:E(G)\to \{1,2,\ldots,|E(G)|\}\), defined by \[f^*(xy)=\left\lceil{\dfrac{|f(x)-f(y)|}{2}}\right\rceil,\] is bijective. The labeling function \(f\) is called an absolute mean graceful labeling of the graph \(G\). In this paper, we obtain absolute mean graceful labelings for \(m\)-splitting and \(m\)-shadow graphs of various graphs.

There are a number of variations of proper edge-colourings of graphs with restrictions on the subgraphs induced by two colour classes. Deciding whether a graph has a 3-edge-colouring with no 2-coloured 4-cycle is NP-complete for cubic graphs with chromatic index 3. But for bipartite cubic graphs the problem is solved completely in this paper. Furthermore, the minimum number of 2-coloured 4-cycles in a 3-edge-colouring is determined for any subcubic bipartite multigraph.

Let \(F_n:=K_1+nK_2\) be a fan of order \(2n+1\). For \(1\le s<t\), we consider the weakened Gallai-Ramsey number \(gr^t_s(F_n)\), defined to be the least \(p\in \mathbb{N}\) such that every Gallai \(t\)-coloring of \(K_p\) contains a subgraph isomorphic to \(F_n\) whose edges use at most \(s\) colors. Our main results include the evaluations \(gr^t_2(F_2)=t+3\), \(gr^3_2(F_3)=9\), and \(gr^t_{2n-1}(F_n)=2n+1\).

We introduce a parity–sum statistic on permutations that assigns to each position a weight determined by the parity of the entry occupying it. When restricted to alternating permutations, this statistic yields two \(q\)–analogues of the Euler numbers, corresponding to the up–down and down–up types. We establish symmetry and reciprocity properties of these polynomials. Specializing at \(q=1\), the resulting recursions reduce to the classical enumerative relations and recover Andr’e’s convolution identity for the Euler numbers. The distribution of the parity–sum statistic over the full symmetric group is also determined.

Previous work by Lesniak (1975) and recently by Qiao and Zhan (2022) established a sharp lower bound for the number of leaves of a tree as a function of its order and diameter. In this note, we derive a lower bound for the number of leaves as a function of the entire sequence of eccentricities, and provide a complete characterisation of all trees attaining equality. We also obtain a new but simpler proof for the diameter-bound. Furthermore, we establish the analogous result for the maximum with respect to the entire eccentric sequence.

Soft set theory, first introduced by Molodtsov, is a flexible approach for handling uncertainty-related problems and modeling uncertain information. Since soft set operations form the core of the theory, their algebraic properties and related structures have attracted considerable research interest. Several forms of symmetric difference operations have been proposed, including restricted and extended symmetric difference operations. Although restricted symmetric difference has already been defined, its definition is not fully consistent with the general formula of restricted soft set operations. In this paper, we first provide an alternative definition of restricted symmetric difference that follows the general form of restricted soft set operations. We then investigate its algebraic properties together with the extended symmetric difference operation, both for soft sets with a fixed parameter set and for soft sets over \(U\). We also establish new properties analogous to the symmetric difference operation in classical set theory, including the case where parameter sets may be disjoint. By deriving distributive rules, we show that important algebraic structures arise when restricted or extended symmetric difference operations are combined with other soft set operations. This study fills a gap in the literature, guides readers new to the theory, and presents a comprehensive analysis of restricted and extended symmetric difference operations, including corrected theorems and proofs from earlier studies.

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.