## DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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DMTCS Conference vol AE (2005), pp. 87-92

## 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

### DMTCS Conference Volume AE (2005), pp. 87-92

author: Martin Charles Golumbic, Marina Lipshteyn and Michal Stern Representations of Edge Intersection Graphs of Paths in a Tree Paths of a tree, Intersection graphs, Weakly chordal graphs, Coloring, EPT-graphs Let P be a collection of nontrivial simple paths in a tree T . The edge intersection graph of P , denoted by EPT( P ) , has vertex set that corresponds to the members of P , and two vertices are joined by an edge if the corresponding members of P share a common edge in T . An undirected graph G is called an edge intersection graph of paths in a tree, if G = EPT( P ) for some P and T . The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. Martin Charles Golumbic and Marina Lipshteyn and Michal Stern (2005), Representations of Edge Intersection Graphs of Paths in a Tree, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 87-92 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAE0118.ps.gz (99 K) dmAE0118.ps (403 K) dmAE0118.pdf (156 K)