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Partition of Sparse Multigraphs into a Forest and a Forest with Restrictions

Ilkyoo Choi, Alexandr V. Kostochka, Matthew Yancey

TL;DR

This work investigates when sparse multigraphs can be partitioned into two forests with additional restrictions, by introducing and analyzing two families of colorings: $(F^Δ_D, F)$-colorings, where one forest has maximum degree at most $D$, and $(F^e_D, F)$-colorings, where the $M$-components have bounded total capacity. The authors develop a weighted potential framework and gap lemmas, proving tight sparsity thresholds for both multigraphs and simple graphs across odd/even $D$, and for each colorings type, with explicit threshold formulas such as $((4D+3)/(2(D+1)), 1/(2(D+1)))$ for multigraphs in the $(F^Δ_D, F)$-case, and $((6D+5)/(3(D+1)), 2/(3(D+1)))$ for simple graphs with $D\ge2$. They also construct sharpness gadgets to demonstrate tightness and provide a detailed outline of the proof strategy via minimal counterexamples and discharging, culminating in exact results for all considered regimes and highlighting open cases like $D=1$ for simple graphs. Overall, the paper offers exact sparsity conditions guaranteeing forest-forest partitions with restrictive components, advancing the theory of sparse graph colorings and vertex arboricity with refined measures. The methods bridge combinatorial constructions, potential-based analysis, and discharging techniques to deliver precise, extremal bounds.

Abstract

The following measure of sparsity of multigraphs refining the maximum average degree: For $a>0$ and an arbitrary real $b$, a multigraph $H$ is \emph{$(a,b)$-sparse} if it is loopless and for every $A\subseteq V(H)$ with $|A|\geq 2$, the induced subgraph $H[A]$ has at most $a|A|+b$ edges. Forests are exactly $(1,-1)$-sparse multigraphs. It is known that the vertex set of any $(2,-1)$-sparse multigraph can be partitioned into two parts each of which induces a forest. For a given parameter $D$ we study for which pairs $(a,b)$ every $(a,b)$-sparse multigraph $G$ admits a vertex partition $(V_1, V_2)$ of $V(G)$ such that $G[V_1]$ and $G[V_2]$ are forests, and in addition either (i) $Δ(G[V_1])\leq D$ or (ii) every component of $G[V_1]$ has at most $D$ edges. We find exact bounds on $a$ and $b$ for both types of problems (i) and (ii). We also consider problems of type (i) in the class of simple graphs and find exact bounds for all $D\geq 2$.

Partition of Sparse Multigraphs into a Forest and a Forest with Restrictions

TL;DR

This work investigates when sparse multigraphs can be partitioned into two forests with additional restrictions, by introducing and analyzing two families of colorings: -colorings, where one forest has maximum degree at most , and -colorings, where the -components have bounded total capacity. The authors develop a weighted potential framework and gap lemmas, proving tight sparsity thresholds for both multigraphs and simple graphs across odd/even , and for each colorings type, with explicit threshold formulas such as for multigraphs in the -case, and for simple graphs with . They also construct sharpness gadgets to demonstrate tightness and provide a detailed outline of the proof strategy via minimal counterexamples and discharging, culminating in exact results for all considered regimes and highlighting open cases like for simple graphs. Overall, the paper offers exact sparsity conditions guaranteeing forest-forest partitions with restrictive components, advancing the theory of sparse graph colorings and vertex arboricity with refined measures. The methods bridge combinatorial constructions, potential-based analysis, and discharging techniques to deliver precise, extremal bounds.

Abstract

The following measure of sparsity of multigraphs refining the maximum average degree: For and an arbitrary real , a multigraph is \emph{-sparse} if it is loopless and for every with , the induced subgraph has at most edges. Forests are exactly -sparse multigraphs. It is known that the vertex set of any -sparse multigraph can be partitioned into two parts each of which induces a forest. For a given parameter we study for which pairs every -sparse multigraph admits a vertex partition of such that and are forests, and in addition either (i) or (ii) every component of has at most edges. We find exact bounds on and for both types of problems (i) and (ii). We also consider problems of type (i) in the class of simple graphs and find exact bounds for all .

Paper Structure

This paper contains 12 sections, 21 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. History and results
  3. Proof ideas and structure of the paper
  4. Constructions
  5. Premise
  6. Gap Lemmas
  7. $\rho^{m, e}$ with even $D$
  8. Common Reducible Configurations
  9. $(\mathcal{F}^{\Delta}_D, \mathcal{F})$-colorings
  10. $(\mathcal{F}^{e}_D, \mathcal{F})$-colorings
  11. Odd $D$
  12. Concluding remarks

Key Result

Theorem 1.1

For each $D\geq 1$, every $\left(\frac{4D+3}{2(D+1)},\frac{1}{2(D+1)}\right)$-sparse multigraph has an $(\mathcal{F}^{\Delta}_D, \mathcal{F})$-coloring.

Figures (3)

  • Figure 1: Sharpness example for \ref{['thm:multi-degreeSc']}.
  • Figure 2: Sharpness example for \ref{['thm:multi-orderSc']}.
  • Figure 3: Sharpness example for \ref{['thm:simple-degreeSc']}.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7: 2022LiWa
  • proof
  • Lemma 3.1
  • proof
  • ...and 29 more