Some Necessary and Sufficient Conditions for Diophantine Graphs

TL;DR

This paper investigates Diophantine graphs, where a labeling $f$ on $n$ vertices satisfies $\gcd(f(u),f(v)) \mid n$ for each edge; it introduces maximal Diophantine graphs $D_n$ and the generalized $\gamma$-labeling framework to study labeling feasibility. It derives exact expressions for the independence number $\alpha(D_n)$, the number of full-degree vertices $F(D_n)$, and the clique number $Cl(D_n)$ using prime-power label structures and the function $\gamma_x(n)$, including two methods to compute $F(D_n)$ via inclusion-exclusion and $\gamma_x(n)$; it also analyzes the minimum-degree vertex and employs degree-sequence majorization to relate Diophantine graphs to $D_n$ via bounds on $|E(G)|$, $F(G)$, and $\delta(G)$. The paper lists six necessary conditions $C_1$–$C_6$ for a graph to admit a Diophantine labeling and illustrates, with concrete examples, that these conditions are necessary but not sufficient. Overall, it provides a number-theoretic framework tying prime-power label structures to key graph invariants, enabling systematic screening of graphs for possible Diophantine labelings.

Abstract

A linear Diophantine equation $ax + by = n$ is solvable if and only if gcd$(a; b)$ divides $n$. A graph $G$ of order $n$ is called Diophantine if there exists a labeling function $f$ of vertices such that gcd$(f(u); f(v))$ divides $n$ for every two adjacent vertices $u; v$ in $G$. In this work, maximal Diophantine graphs on $n$ vertices, $D_n$, are defined, studied and generalized. The independence number, the number of vertices with full degree and the clique number of $D_n$ are computed. Each of these quantities is the basis of a necessary condition for the existence of such a labeling.

Figures (5)

• Figure 1: Some maximal Diophantine graphs $D_9$, $D_{10}$ and $D_{11}$
• Figure 2: $(\Gamma_n,f_1)\cong_l (\acute{\Gamma}_n,f_2)$
• Figure 3:
• Figure 4: The graphs $G_1,G_2,G_3,G_4,G_5$ and $G_6$ are non-Diophantine
• Figure 5: A graph $G$ does not satisfy the six necessary conditions thought $G$ is not Diophantine

Theorems & Definitions (47)

• Definition 1.1
• Example 1.1
• Definition 1.2
• Lemma 1.1
• Theorem 1.1
• Lemma 1.2
• Theorem 1.2
• Corollary 1.1
• Definition 2.1
• Definition 2.2