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Graphs and Combinatorics

, Volume 35, Issue 1, pp 91–102 | Cite as

Graph 2-rankings

  • Jordan Almeter
  • Samet Demircan
  • Andrew Kallmeyer
  • Kevin G. MilansEmail author
  • Robert Winslow
Original Paper
  • 18 Downloads

Abstract

A 2-ranking of a graph G is an ordered partition of the vertices of G into independent sets \(V_1, \ldots , V_t\) such that for \(i<j\), the subgraph of G induced by \(V_i \cup V_j\) is a star forest in which each vertex in \(V_i\) has degree at most 1. A 2-ranking is intermediate in strength between a star coloring and a distance-2 coloring. The 2-ranking number ofG, denoted \(\chi _{2}(G)\), is the minimum number of parts needed for a 2-ranking. For the d-dimensional cube \(Q_d\), we prove that \(\chi _{2}(Q_d) = d+1\). As a corollary, we improve the upper bound on the star chromatic number of products of cycles when each cycle has length divisible by 4. Let \(\chi _{2}'(G)=\chi _{2}(L(G))\), where L(G) is the line graph of G; equivalently, \(\chi _{2}'(G)\) is the minimum t such that there is an ordered partition of E(G) into t matchings \(M_1, \ldots , M_t\) such that for each j, the matching \(M_j\) is induced in the subgraph of G with edge set \(M_1 \cup \cdots \cup M_j\). We show that \(\chi _{2}'(K_{m,n})=nH_m\) when m! divides n, where \(K_{m,n}\) is the complete bipartite graph with parts of sizes m and n, and \(H_m\) is the harmonic sum \(1 + \cdots + \frac{1}{m}\). We also prove that \(\chi _{2}(G) \le 7\) when G is subcubic and show the existence of a graph G with maximum degree k and \(\chi _{2}(G) \ge \varOmega (k^2/\log (k))\).

Keywords

Graph ranking Star coloring 

Notes

Acknowledgements

This research was supported in part by NSA Grant H98230-14-1-0325.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of William and MaryWilliamsburgUSA
  2. 2.Stony Brook UniversityStony BrookUSA
  3. 3.Miami UniversityOxfordUSA
  4. 4.University of KansasLawrenceUSA

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