Research Article
Results on Grundy Edge Coloring of Some Special Graphs
Issue:
Volume 14, Issue 6, December 2026
Pages:
389-400
Received:
6 September 2026
Accepted:
29 September 2026
Published:
4 November 2026
Abstract: An edge coloring of a simple, connected, undirected graph G is an assignment of color labels to the edges of G such that no two adjacent edges are assigned the same color. The first-fit or greedy edge coloring algorithm colors edges sequentially by assigning the smallest available positive integer color to each edge. The Grundy edge chromatic number, denoted by Γ'(G), captures the worst-case performance of this greedy approach, defined formally as the maximum number of colors utilized by the greedy edge coloring algorithm across all possible orderings of the edge set E(G). Determining Γ'(G) is known to be computationally hard in general, prompting significant research into deriving exact values and structural bounds for specialized graph families and graph operations. In this paper, we explore the Grundy edge chromatic number for several key topological families and structured composite graphs. First, we focus on the shadow graph operation, denoted D2(G), which duplicates the vertex set of G and interconnects vertices based on their neighborhood adjacencies. We establish exact explicit formulas for the Grundy edge chromatic number of shadow graphs constructed from foundational graph structures, including paths Pn, cycles Cn, sunlet graphs Sn, and tadpole graphs Tn,k. Second, we extend our investigation to complex wheel-like and symmetric network architectures. Specifically, we evaluate the structural properties and derive exact closed-form expressions for the Grundy edge chromatic number of closed helm graphs CHn, double wheel graphs DWn, and djembe graphs Dn. For each considered graph class, we construct explicit edge orderings that force the greedy algorithm to attain its theoretical upper bound, thereby matching the structural upper bounds derived from degree conditions and neighborhood characteristics. These exact results provide comprehensive insights into the performance bounds of greedy edge-labeling heuristics in interconnection networks, expanding the current understanding of greedy coloring parameters in graph theory.
Abstract: An edge coloring of a simple, connected, undirected graph G is an assignment of color labels to the edges of G such that no two adjacent edges are assigned the same color. The first-fit or greedy edge coloring algorithm colors edges sequentially by assigning the smallest available positive integer color to each edge. The Grundy edge chromatic numbe...
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Research Article
Arithmetic Pencils and Rank Two Congruent Numbers Elliptic Surfaces via Geometric Sections
Kouakou Kouassi Vincent*,
Edjabrou Ulrich Blanchard Kablam
Issue:
Volume 14, Issue 6, December 2026
Pages:
401-412
Received:
16 September 2026
Accepted:
9 October 2026
Published:
10 November 2026
DOI:
10.11648/j.ajam.20261406.12
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Abstract: We investigates an arithmetic family of congruent-number elliptic curves arising from the Euclid parametrization of Pythagorean triples, with the aim of determining its generic Mordell-Weil rank and understanding the occurrence of higher ranks under specialization. Our approach combines explicit constructions of rational sections with the geometric theory of elliptic surfaces. A suitable quadratic base change produces a new elliptic surface on which two independent rational sections can be exhibited explicitly. The singular fibres are classified using Kodaira's theory, and the associated global invariants are computed to obtain a geometric upper bound for the Mordell–Weil rank. The Shioda–Tate formula, together with the complex multiplication structure and Galois descent, is then used to establish the exact generic rank over the rational function field. We also examine the geometric origin of a natural non-torsion section arising from an arithmetic pencil of lines and relate its coordinates to a fixed auxiliary cubic curve. Finally, we study the arithmetic behavior of specialized fibres through computational searches and rigorous rank certificates based on rational points, height-pairing matrices, and descent bounds. The results distinguish the generic rank from the higher ranks occurring at particular parameter values and reveal explicit specializations with rank at least three, including examples with rank at least four. We also establish a connection with a previously known rank-three family, interpreting it through a quadratic extension and the decomposition into invariant and anti-invariant sections. These findings illustrate how explicit arithmetic constructions and the geometry of elliptic surfaces can work together to clarify generic rank and rank-jump phenomena in families of congruent-number elliptic curves. We further relate our construction to the rank-three families of Halbeisen, Hungerbühler, and Shamsi Zargar, interpreting their arithmetic through a quadratic extension and the decomposition into invariant and anti-invariant sections. This connection illustrates how dilations of classical congruent numbers (the Euclid areas), explicit arithmetic constructions, and the geometry of elliptic surfaces contribute to the study of generic ranks and rank jumps in families of congruent-number elliptic curves.
Abstract: We investigates an arithmetic family of congruent-number elliptic curves arising from the Euclid parametrization of Pythagorean triples, with the aim of determining its generic Mordell-Weil rank and understanding the occurrence of higher ranks under specialization. Our approach combines explicit constructions of rational sections with the geometric...
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,
Ponpandi Periasamy
