Edge Intersection Graphs of Paths on a Triangular Grid

TL;DR

This paper introduces and studies a new class of intersection graphs called EPGt graphs, focusing on their properties, representations, and coloring characteristics, especially for graphs with limited bends.

Contribution

The paper defines B_k-EPGt graphs, explores their properties, characterizes certain subgraph representations, and establishes bounds on Helly number and clique coloring.

Findings

B_{2}-EPG graphs can be B_{1}-EPGt.

Characterization of clique and cycle representations in B_{1}-EPGt.

B_{1}-EPGt graphs have Strong Helly number 3 and are 7-clique colorable.

Abstract

We introduce a new class of intersection graphs, the edge intersection graphs of paths on a triangular grid, called EPGt graphs. We show similarities and differences from this new class to the well-known class of EPG graphs. A turn of a path at a grid point is called a bend. An EPGt representation in which every path has at most $k$ bends is called a B$_k$-EPGt representation and the corresponding graphs are called B$_k$-EPGt graphs. We provide examples of B$_{2}$-EPG graphs that are B$_{1}$-EPGt. We characterize the representation of cliques with three vertices and chordless 4-cycles in B$_{1}$-EPGt representations. We also prove that B$_{1}$-EPGt graphs have Strong Helly number $3$. Furthermore, we prove that B$_{1}$-EPGt graphs are $7$-clique colorable.