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Coloring Graphs to Produce Properly Colored Walks

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Abstract

For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same color. We show that the proper-walk connection number is at most three for all cyclic graphs, and at most two for bridgeless graphs. We also characterize the bipartite graphs that have proper-walk connection number equal to two, and show that this characterization also holds for the analogous problem where one is restricted to properly colored paths.

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Acknowledgements

We thanks the referees for their thoughtful comments that improved the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, Clemson, USA

    Robert Melville & Wayne Goddard

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  1. Robert Melville
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  2. Wayne Goddard
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Correspondence to Wayne Goddard.

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Melville, R., Goddard, W. Coloring Graphs to Produce Properly Colored Walks. Graphs and Combinatorics 33, 1271–1281 (2017). https://doi.org/10.1007/s00373-017-1843-y

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  • DOI: https://doi.org/10.1007/s00373-017-1843-y

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