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\begin{document}

\title{Colored Pentagon Reductions}

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\author{Eric Bauerfeld}
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\subjclass[2010]{Primary }

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\dedicatory{}

\begin{abstract}
  We investigate specific types of minors we can obtain by a sequence of contractions of two incident to a sequence of vertices of degree 5 relative to a proper coloring.
\end{abstract}

\maketitle

\section{Definitions}

\begin{definition}
Let $G$ be a maximal planar graph with a proper 4-coloring $c : V(G) \to \{1,2,3,4\}$. A \emph{colored pentagon reduction} is a function
\[
  R : (G, c, v_0, v_1, v_2) \mapsto G / \{v_1 v_0,\, v_2 v_0\}
\]
where $v_0 \in V(G)$ has degree 5, $v_1$ and $v_2$ are adjacent vertices satisfying $c(v_1) = c(v_2)$, and $G / \{v_1 v_0,\, v_2 v_0\}$ denotes the minor obtained by contracting $v_1$ and $v_2$ each into $v_0$.
\end{definition}

\begin{lemma}
  If $G' = R(G, c, v_0, v_1, v_2)$, then $G'$ has a proper 4-coloring.
\end{lemma}

\begin{proof}
The idea is to recolor $v_0$ with the shared color of $v_1$ and $v_2$, and check that this creates no new conflicts.

Let $\alpha = c(v_1) = c(v_2)$ and define $c' : V(G') \to \{1,2,3,4\}$ by
\[
  c'(v) = \begin{cases} \alpha & \text{if } v = v_0, \\ c(v) & \text{otherwise.} \end{cases}
\]
Every edge of $G'$ not incident to $v_0$ is an edge of $G$ between vertices outside $\{v_1, v_2\}$, so those edges are still properly colored. It remains to check edges $v_0 w$ in $G'$, i.e., that no neighbor of $v_0$ in $G'$ also has color $\alpha$.

Since $c(v_1) = c(v_2)$ and $c$ is proper, $v_1$ and $v_2$ cannot be adjacent in $G$, so they are non-adjacent in the 5-cycle induced by $N_G(v_0)$ (which exists because $G$ is maximal planar). One can check that in any 5-cycle, two non-adjacent vertices together cover every other vertex --- that is, every remaining vertex is adjacent to at least one of them. Therefore every $w \in N_G(v_0) \setminus \{v_1, v_2\}$ is adjacent in $G$ to $v_1$ or $v_2$, so $c(w) \neq \alpha = c'(v_0)$.

For new neighbors of $v_0$ coming from the contraction, namely $w \in (N_G(v_1) \cup N_G(v_2)) \setminus \{v_0, v_1, v_2\}$, we get $c(w) \neq \alpha = c'(v_0)$ for free since $c$ is proper on $G$.

Hence $c'$ is a proper 4-coloring of $G'$.
\end{proof}

\end{document}

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