We study edge partitions of a bipartite graph into induced-2K₂-free bipartite graphs, i.e. into Ferrers (chain) graphs. We define fp(G) as the minimum number of parts in such a partition. We prove general lower and upper bounds in terms of induced matchings and Dilworth widths of neighborhood posets. We compute the parameter exactly for paths and even cycles, and we exhibit separations showing that the induced-matching lower bound and the width upper bound can both be far from tight. We also record a simple host-induced conflict-graph lower bound, present a 0–1 matrix viewpoint, and add some complexity remarks.

We study the Equivalent Local Sequence Problem (ELSP), which consists in recovering an explicit sequence of local complementations that transforms a graph into a locally equivalent one. Focusing on directed Paley graphs, we establish that local complementations commute and induce a free action of an elementary abelian 2-group. The stabilizer condition is reformulated as a system of convolution equations and analyzed through Fourier techniques over finite fields, leading to a proof of stabilizer triviality. As a consequence, each graph in the orbit admits a unique subset encoding, and ELSP reduces to solving a linear inversion problem over 𝔽₂. This characterization completely resolves ELSP for directed Paley graphs, provides a polynomial-time inversion algorithm and highlights structural features that may support future developments in cryptographic frameworks and quantum graph-state models.

Given a configuration of pebbles on the edges of a connected graph G, an edge pebbling move is defined as the removal of two pebbles off an edge and placing one on an adjacent edge. The domination cover edge pebbling number of a graph G is the minimum number of pebbles required such that the set of edges that contain pebbles form an edge dominating set S of G, for the initial configuration of pebbles can be altered by a sequence of pebbling moves and it is denoted by ψ_e(G) for a graph G. In this paper, we determine ψ_e(G) for Generalized Petersen graph, Jewel graph and Triangular snake graph.

In this paper we study group divisible designs (GDDs) with block size 4 and two groups of different sizes when λ₂ = 1. We obtain necessary conditions for the existence of such GDDs and prove that these necessary conditions are sufficient in several cases. Further, we present general constructions using resolvable designs.

In 1940, Birkhoff posed an open problem of counting all finite lattices on n elements. Recently, Bhavale counted all non-isomorphic lattices on n elements, containing up to four reducible elements, and having nullity up to three. Further, Aware and Bhavale counted all non-isomorphic lattices on n elements, containing up to five comparable reducible elements, and having nullity up to three. In this paper, we count all non-isomorphic lattices on n elements, containing five reducible elements, and having nullity three.

The unit graph of a commutative ring with a non-zero identity is a graph with vertices as ring elements, and there is an edge between two distinct vertices if their sum is a unit. This study investigates the decomposition of the unit graph by examining its induced subgraphs and analyze key graph invariants, such as connectivity, diameter, and girth, for a finite local ring. We further decompose the unit graph of certain finite commutative rings into fundamental structures, such as cycle and star graphs.