Coloring Grids Avoiding Bicolored Paths

TL;DR

The paper studies $P_k$-coloring on grid graphs, where a proper coloring avoids a bicolored $P_k$ and is quantified by $s_k(G)$. It introduces a detailed boundary-structure analysis of bicolored components, including peripheral vs partial components and a boundary walk $B_C$, supported by a boundary-angle lemma. The main result shows that for every $k\ge5$ and $m,n\ge k-2$, $s_k(P_m\square P_n)=4$, matching the known upper bound and thus settling the grid case; it also notes 3-color constructions that succeed when one dimension is smaller than $k-2$. This work confirms a constant gap between the ordinary chromatic number and the $P_k$-chromatic number on 2D grids and complements related results on nonrepetitive and acyclic colorings.

Abstract

The star chromatic number on a graph is the minimum number of colors in a proper vertex coloring forbidding any $P_4$ with two colors (bicolored). This problem was introduced by Grünbaum (1973) together with the acyclic coloring of graphs, where bicolored cycles are avoided. In this paper, we study a generalization of this problem, by considering proper vertex coloring on graphs forbidding bicolored paths of a fixed length, which was initially discussed by Alon, McDiarmid, and Reed (1991). Here, we study this problem on products of two paths. We show that at least 4 colors are needed to properly color the product of paths, $P_m\square P_n$, avoiding a bicolored $P_k,$ unless $n<k-2$ or $m<k-2.$ With this result, the above question is settled for all $k$ on 2-dimensional grids.

Figures (4)

• Figure 1: In (a)-(c), examples of bicolored components show the edges of $B_C$ (bold edges) and the neighboring edges of another bicolored component (dashed edges). In (d), a coloring of the grid shows examples of peripheral bicolored components.
• Figure 2: Invalid cases listing values of $(\beta,\gamma)$.
• Figure 3: The only valid configurations for $(\beta,\gamma)$ (omitting symmetric cases), except the leftmost case. The vertices marked with * are part of $C.$
• Figure 4: The neighboring columns of maximal red-blue colored $P$ (bold edges) and possible colorings: (a) $i \not\equiv k$ (mod 2) and $i$, even, (b) $i \equiv k$ (mod 2) and $i$, even, (c) $i \not\equiv k$ (mod 2) and i, odd, (d) $i \equiv k$ (mod 2) and $i$, odd.

Theorems & Definitions (12)

• Theorem 1
• Definition 1
• Definition 2
• Definition 3
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 4
• proof