Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrepancies of spanning trees in dense graphs

Lawrence Hollom, Lyuben Lichev, Adva Mond, Julien Portier

Abstract

We address several related problems on combinatorial discrepancy of trees in a setting introduced by Erdős, Füredi, Loebl and Sós. Given a fixed tree $T$ on $n$ vertices and an edge-colouring of the complete graph $K_n$, for every colour, we find a copy of $T$ in $K_n$ where the number of edges in that colour significantly exceeds its expected count in a uniformly random embedding. This resolves a problem posed by Erdős, Füredi, Loebl and Sós by generalising their work from two to many colours. Furthermore, if $T$ has maximum degree $Δ\leqεn$ for sufficiently small $ε> 0$ and the edge-colouring of $K_n$ is both balanced and ``not too close'' to one particular instance, we show that, for every colour, there is a copy of $T$ in $K_n$ where that colour appears on linearly more edges than any other colour. Several related examples are provided to demonstrate the necessity of the introduced structural restrictions. Our proofs combine saturation arguments for the existence of particular coloured substructures and analysis of conveniently defined local exchanges. Using similar methods, we investigate the existence of copies of a graph $H$ with prescribed number of edges in each colour in $2$-edge-coloured dense host graphs. In particular, for a graph $H$ with bounded maximum degree and balanced $2$-edge-colourings $\mathbf{c}$ of a host graph $G$ with minimum degree at least $(1-ε)n$ for some $ε> 0$, we show that, for any sufficiently large $n$ and sufficiently small $ε$, there exists a copy of $H$ where the number of edges in the two colours differ by at most $2$. Moreover, we completely characterise the pairs $(H,\mathbf{c})$ for which the difference of $2$ cannot be improved, refuting a conjecture by Mohr, Pardey, and Rautenbach.

Discrepancies of spanning trees in dense graphs

Abstract

We address several related problems on combinatorial discrepancy of trees in a setting introduced by Erdős, Füredi, Loebl and Sós. Given a fixed tree on vertices and an edge-colouring of the complete graph , for every colour, we find a copy of in where the number of edges in that colour significantly exceeds its expected count in a uniformly random embedding. This resolves a problem posed by Erdős, Füredi, Loebl and Sós by generalising their work from two to many colours. Furthermore, if has maximum degree for sufficiently small and the edge-colouring of is both balanced and ``not too close'' to one particular instance, we show that, for every colour, there is a copy of in where that colour appears on linearly more edges than any other colour. Several related examples are provided to demonstrate the necessity of the introduced structural restrictions. Our proofs combine saturation arguments for the existence of particular coloured substructures and analysis of conveniently defined local exchanges. Using similar methods, we investigate the existence of copies of a graph with prescribed number of edges in each colour in -edge-coloured dense host graphs. In particular, for a graph with bounded maximum degree and balanced -edge-colourings of a host graph with minimum degree at least for some , we show that, for any sufficiently large and sufficiently small , there exists a copy of where the number of edges in the two colours differ by at most . Moreover, we completely characterise the pairs for which the difference of cannot be improved, refuting a conjecture by Mohr, Pardey, and Rautenbach.

Paper Structure

This paper contains 25 sections, 19 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Maximising the number of edges in some colour
  3. Maximising the number of edges in a given colour
  4. k-sum subgraphs of dense 2-edge-coloured graphs
  5. Brief outline of the proofs.
  6. Plan of the paper.
  7. Preliminary results and tools
  8. Switchable edges and the switching technique
  9. Case 1: $d_1\ge 2\zeta - 1 + \Delta$.
  10. Case 2: $d_1 \le 2\zeta - 2 + \Delta$.
  11. Case 2.1: $\sum_{i=1}^{\infty} \ell_i \ge 4\zeta-2$.
  12. Case 2.2: $\sum_{i=1}^{\infty} \ell_i\le 4\zeta -3$.
  13. Marked 4-cycles
  14. 4-cycles in edge-coloured graphs
  15. Nearly balanced r-edge-colourings of K_n
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exists a universal constant $c>0$ such that the following holds. Fix an integer $r\ge 2$ and an $r$-colouring of the edges of $K_n$. Then, for every tree $T$ on $n$ vertices with maximum degree $\Delta$, there exists a colour $i\in [r]$ and a copy of $T$ in which the number of edges in colour

Figures (2)

  • Figure 1: An illustration of a switchable pair of edges $uw, vz$ of type I, II and III, respectively. The vertices marked with small circles have all their neighbours in the figure.
  • Figure 2: The three cases of a marked 4-cycle. Solid edges are blue, and dashed edges are some colour other than blue.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Example 1.5
  • Definition 1.6
  • Theorem 1.7
  • Example 1.8
  • Theorem 1.9
  • Example 1.10
  • ...and 40 more