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Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 373–388 | Cite as

On vertex-parity edge-colorings

  • Borut Lužar
  • Mirko Petruševski
  • Riste Škrekovski
Article

Abstract

A vertex signature \(\pi \) of a finite graph G is any mapping \(\pi \,{:}\,V(G)\rightarrow \{0,1\}\). An edge-coloring of G is said to be vertex-parity for the pair \((G,\pi )\) if for every vertex v each color used on the edges incident to v appears in parity accordance with \(\pi \), i.e. an even or odd number of times depending on whether \(\pi (v)\) equals 0 or 1, respectively. The minimum number of colors for which \((G,\pi )\) admits such an edge-coloring is denoted by \(\chi '_p(G,\pi )\). We characterize the existence and prove that \(\chi '_p(G,\pi )\) is at most 6. Furthermore, we give a structural characterization of the pairs \((G,\pi )\) for which \(\chi '_p(G,\pi )=5\) and \(\chi '_p(G,\pi )=6\). In the last part of the paper, we consider a weaker version of the coloring, where it suffices that at every vertex, at least one color appears in parity accordance with \(\pi \). We show that the corresponding chromatic index is at most 3 and give a complete characterization for it.

Keywords

Vertex-parity edge-coloring Vertex-parity chromatic index Weak vertex-parity edge-coloring Vertex signature 

Notes

Acknowledgements

This work is partially supported by Slovenian Research Agency Program P1-0383.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Information StudiesNovo MestoSlovenia
  2. 2.Department of Mathematics and InformaticsFaculty of Mechanical Engineering-SkopjeSkopjeRepublic of Macedonia
  3. 3.Institute of MathematicsPhysics and MechanicsLjubljanaSlovenia
  4. 4.University of Primorska, FAMNITKoperSlovenia

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