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Eigenvalue bounds for distance-edge colorings

Aida Abiad, Harper Reijnders

Abstract

For a fixed positive integer $t$, we consider the graph colouring problem in which edges at distance at most $t$ are given distinct colours. We obtain sharp lower bounds for the distance-$t$ chromatic index, the least number of colours necessary for such a colouring. Our bounds are of algebraic nature; they depend on the eigenvalues of the line graph and on a polynomial which can be found using integer linear programming methods. We show several graph classes that attain equality for our bounds, and also present some computational results which illustrate the bound's performance. Lastly, we investigate the implications the spectral approach has to the Erdős-Nešetřil conjecture, and derive some conditions which a graph must satisfy if we could use it to obtain a counter example through the proposed spectral methods.

Eigenvalue bounds for distance-edge colorings

Abstract

For a fixed positive integer , we consider the graph colouring problem in which edges at distance at most are given distinct colours. We obtain sharp lower bounds for the distance- chromatic index, the least number of colours necessary for such a colouring. Our bounds are of algebraic nature; they depend on the eigenvalues of the line graph and on a polynomial which can be found using integer linear programming methods. We show several graph classes that attain equality for our bounds, and also present some computational results which illustrate the bound's performance. Lastly, we investigate the implications the spectral approach has to the Erdős-Nešetřil conjecture, and derive some conditions which a graph must satisfy if we could use it to obtain a counter example through the proposed spectral methods.

Paper Structure

This paper contains 13 sections, 25 theorems, 43 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectral lower bounds
  4. Optimization of the bounds on chit'(G)
  5. Closed formulas for the bounds on chi'2(G) and chi'3(G)
  6. Tightness and performance of the lower bounds
  7. Balanced bipartite product
  8. An infinite family of graphs for which Corollary \ref{['cor:ratio_chiprime_2_reg']} is tight
  9. An infinite family of graphs for which Corollary \ref{['cor:ratio_chiprime_3']} is tight
  10. Computational performance
  11. On the distance-t Erdős-Nešetřil problem
  12. Wilf's bound and the Erdős-Nešetřil conjecture
  13. Hoffman-type bound and the Erdős-Nešetřil conjecture

Key Result

Lemma 2

(biggs_algebraic_1974) Let $G$ be a $k$-regular graph, then $L(G)$ is $(2k-2)$-regular.

Figures (5)

  • Figure 1: The balanced bipartite product $C_4 \bowtie C_4 = Q_3$.
  • Figure 2: An example of how to $6$-colour part of $C_q \bowtie C_{q'}$ when $q,q'$ are multiples of $4$. The square nodes represent the remainder of the graph
  • Figure 3: An example of how to $12$-colour part of $GM(2k)$. Dashed and non-dashed lines represent different colours. The square nodes represent the remainder of the graph.
  • Figure 4: The graph $G_3$ as defined in the proof of Proposition \ref{['prop:t-crit']}.
  • Figure 5: A visual proof of why $G_2$ as defined in the proof of Proposition \ref{['prop:t-crit']} is $2$-critical.

Theorems & Definitions (51)

  • Conjecture 1
  • Lemma 2
  • Corollary 3
  • Theorem 4: First inertial-type bound
  • proof
  • Theorem 5: Second inertial-type bound
  • proof
  • Theorem 6: General Hoffman-type bound
  • proof
  • Theorem 7: Hoffman-type bound
  • ...and 41 more