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A note on extendable sets of colorings and rooted minors

Zdeněk Dvořák, Jan M. Swart

TL;DR

The paper addresses the problem of extending $k$-colorings of a vertex set $X$ within graphs while controlling both $X$-rooted and global minor constraints. It generalizes the DeVos–Seymour result from $k=3$ to all $k\nge 3$ by constructing a graph $G$ that realizes any given set $C$ of $k$-colorings of $X$ and is simultaneously $X$-rooted-$K_{k+1}$-minor-free and $K_{k+2}$-minor-free. The proof builds two gadgets, $F_{ ext{copy},k}$ and $F_{ ext{enc},s,k}$, with their $+$ versions being $K_{k+2}$-minor-free, and uses them to encode and propagate color information through pillars $G_1$ and $G_2$ that glue into a final graph $G$ via clique-sums. This approach shows that reducible configurations cannot exploit global properties beyond the $K_{k+2}$-minor-free bound for the range of colors considered, clarifying the limits of Kempe-based reducibility in minor-closed coloring problems and aligning with Hadwiger-type questions on colorable restricted graphs.

Abstract

DeVos and Seymour (2003) proved that for every set $C$ of 3-colorings of a set $X$ of vertices, there exists a plane graph $G$ with vertices of $X$ incident with the outer face such that a 3-coloring of $X$ extends to a 3-coloring of $G$ if and only if it belongs to $C$. We prove a generalization of this claim for $k$-colorings of $X$-rooted-$K_{k+1}$-minor-free $K_{k+2}$-minor-free graphs.

A note on extendable sets of colorings and rooted minors

TL;DR

The paper addresses the problem of extending -colorings of a vertex set within graphs while controlling both -rooted and global minor constraints. It generalizes the DeVos–Seymour result from to all by constructing a graph that realizes any given set of -colorings of and is simultaneously -rooted--minor-free and -minor-free. The proof builds two gadgets, and , with their versions being -minor-free, and uses them to encode and propagate color information through pillars and that glue into a final graph via clique-sums. This approach shows that reducible configurations cannot exploit global properties beyond the -minor-free bound for the range of colors considered, clarifying the limits of Kempe-based reducibility in minor-closed coloring problems and aligning with Hadwiger-type questions on colorable restricted graphs.

Abstract

DeVos and Seymour (2003) proved that for every set of 3-colorings of a set of vertices, there exists a plane graph with vertices of incident with the outer face such that a 3-coloring of extends to a 3-coloring of if and only if it belongs to . We prove a generalization of this claim for -colorings of -rooted--minor-free -minor-free graphs.

Paper Structure

This paper contains 2 sections, 3 theorems, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. Proof

Key Result

Theorem 1

For every cyclically ordered set $X$ of vertices, every set of $3$-colorings of $X$ is planarly realizable.

Figures (2)

  • Figure 1: Pieces of the gadget $F_{\mathrm{enc},4,5}(u,v,w;y_4,y_5)$.
  • Figure 2: The construction from the proof of Theorem \ref{['thm-main']}, the case $m=2$ and $k=6$. The apex vertices $y_4$, …, $y_k$ are not shown.

Theorems & Definitions (6)

  • Theorem 1: DeVos and Seymour DS03
  • Corollary 2
  • Theorem 3
  • Conjecture 4
  • proof
  • proof : Proof of Theorem \ref{['thm-main']}