On Geodesic Leech Labeling of Some Graph Classes

TL;DR

The paper studies geodesic Leech labeling, where an edge labeling induces geodesic path weights that are exactly $\{1,2,...,t_{gp}(G)\}$. It develops a key counting identity for edge-transitive graphs and applies it to families such as cycles $C_n$, complete bipartite graphs $K_{n,n}$, and wheel graphs $W_n$, deriving new constraints and exact results. It proves that cycles $C_n$ with $n\ge5$ are not geodesic Leech, shows at most three regular complete bipartite graphs can be geodesic Leech (specifically $K_{1,1}$, $K_{2,2}$, and $K_{5,5}$), and determines the geodesic path number for wheels $W_n$ with explicit labelings for $W_5$ and $W_6$ while leaving $W_n$ for $n\ge7$ open. The results refine the landscape of geodesic Leech graphs, identify non-characterization by degree sequence, and guide future exploration of this labeling paradigm in graph families. Throughout, the work combines combinatorial counting with targeted analyses and computational checks (e.g., for cycles and wheel graphs) to map feasible graphs admitting geodesic Leech labelings.

Abstract

Let $f:E\rightarrow \{1,2,3,\dots\}$ be an edge labeling of $G$. The geodesic path number of $G$, $t_{gp}(G)$, is the number of geodesic paths in $G$. An edge labeling $f$ is called a geodesic Leech labeling, if the set of weights of the geodesic paths in $G$ is $\{1,2,3,\dots,t_{gp}(G)\}$, where the weight of a path $P$ is the sum of the labels assigned to the edges of $P$. A graph which admits a geodesic Leech labeling is called a geodesic Leech graph. Otherwise, we call it a non-geodesic Leech graph. In this paper, we prove that cycles $C_n$, $n \geq 5$ are non-geodesic Leech graphs. We also prove that there are at most three regular complete bipartite graphs that are geodesic Leech. We show that degree sequence cannot characterize geodesic Leech graphs. The geodesic path number of the wheel graph $W_n$ is obtained and the geodesic Leech labeling of $W_5$ and $W_6$ is given.

Figures (6)

• Figure 1: Geodesic Leech labeling of $C_3$ and $C_4$
• Figure 2: Geodesic Leech Cubic graph on 6 vertices
• Figure 3: Geodesic Leech labeling of $W_5$ and $W_6$
• Figure 4: Almost Geodesic Leech Labeling of $W_7$
• Figure 5: Geodesic Leech labeling of 8 Forbidden Subgraphs of Linegraphs

Theorems & Definitions (13)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Theorem 2.5
• Lemma 2.6
• Theorem 2.7
• Lemma 2.8
• proof