On Zero-Error Capacity of Graphs with One Edge

TL;DR

This paper investigates the zero-error capacity of graphs with a single edge, providing a construction method that achieves capacity in most cases, especially for binary channels with memory, and classifies these graphs into categories based on symmetry.

Contribution

The paper introduces a new code construction method for graphs with one edge, establishing lower bounds and achieving zero-error capacity in most cases.

Findings

Code achieves zero-error capacity for most one-edge graphs

Classification of 28 graphs into 11 symmetry-based categories

Identifies exceptions where the code does not achieve capacity

Abstract

In this paper, we study the zero-error capacity of channels with memory, which are represented by graphs. We provide a method to construct code for any graph with one edge, thereby determining a lower bound on its zero-error capacity. Moreover, this code can achieve zero-error capacity when the symbols in a vertex with degree one are the same. We further apply our method to the one-edge graphs representing the binary channels with two memories. There are 28 possible graphs, which can be organized into 11 categories based on their symmetries. The code constructed by our method is proved to achieve the zero-error capacity for all these graphs except for the two graphs in Case 11.