## 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.

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## References

Akiyama J, Kano M (2011) Factors and factorizations of graphs: proof techniques in factor theory. Springer, New York

Atanasov R, Petruševski M, Škrekovski R (2016) Odd edge-colorability of subcubic graphs. Ars Math Contemp 10(2):359–370

Bondy JA, Murty USR (2008) Graph theory, graduate texts in mathematics, vol 244. Springer, New York

Bondy JA, Murty USR (1976) Graph theory with applications. Elsevier, North-Holland

Czap J, Jendrol’ S, Kardoš F’, Soták R (2012) Facial parity edge colouring of plane pseudographs. Discrete Math 312:2735–2740

Jaeger F (1979) Flows and generalized coloring theorems in graphs. J Combin Theory Ser B 26:205–216

Lužar B, Petruševski M, Škrekovski R (2015) Odd edge coloring of graphs. Ars Math Contemp 9:277–287

Lužar B, Škrekovski R (2013) Improved bound on facial parity edge coloring. Discrete Math 313:2218–2222

Lovász L (1972) The factorization of graphs II. Acta Math Acad Sci Hungar 23:465–478

Petruševski M (2015) A note on weak odd edge colorings of graphs. Adv Math Sci J 4:7–10

Petruševski M (2017) Odd \(4\)-edge-colorability of graphs. J Graph Theory. doi:10.1002/jgt.22168

Pyber L (1991) Covering the edges of a graph by..., sets, graphs and numbers. Colloq Math Soc János Bolyai 60:583–610

Schrijver A (2003) Combinatorial optimization. Polyhedra and efficiency, Vol A: Algorithms and combinatorics. Springer, Berlin

Seymour PD (1979) Sums of circuits. In: Bondy A, Murty USR (eds) Graph theory and related topics. Academic Press, New York, pp 342–355

Shu J, Zhang C-Q, Zhang T (2012) Flows and parity subgraphs of graphs with large odd-edge-connectivity. J Combin Theory Ser B 102:839–851

Szabó J (2006) Graph packings and the degree prescribed subgraph problem, Eötvös Loránd University. Doctoral thesis

Szekeres G (1973) Polyhedral decompositions of cubic graphs. Bull Austral Math Soc 8:367–387

West D B (2001) Introduction to graph theory. Pearson Education, London

Yu Q R, Liu G (2009) Graph factors and matching extensions. Springer, Berlin

## Acknowledgements

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

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Lužar, B., Petruševski, M. & Škrekovski, R. On vertex-parity edge-colorings.
*J Comb Optim* **35, **373–388 (2018). https://doi.org/10.1007/s10878-017-0178-1

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### Keywords

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