Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

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 $$\hbox {B}_k$$ B k -EPGt representation and the corresponding graphs are called $$\hbox {B}_k$$ B k -EPGt graphs. We provide examples of $$\hbox {B}_{{2}}$$ B 2 -EPG graphs that are $$\hbox {B}_{{1}}$$ B 1 -EPGt. We characterize the representation of cliques with three vertices and chordless 4-cycles in $$\hbox {B}_{{1}}$$ B 1 -EPGt representations. We also prove that $$\hbox {B}_{{1}}$$ B 1 -EPGt graphs have Strong Helly number 3. Furthermore, we prove that $$\hbox {B}_{{1}}$$ B 1 -EPGt graphs are 7-clique colorable.