Packing Chromatic Number of Caterpillars

• The paper’s main contribution classifies caterpillars with a packing chromatic number of 3 using periodic backbone color patterns and explicit structural families.
• It details structural conditions and pendant-placement constraints that yield tight upper bounds (6 for shorter backbones, 7 in general) for these graphs.
• The study also highlights open algorithmic challenges, notably the complexity of recognizing caterpillars with a packing chromatic number of 4.

The packing chromatic number of a caterpillar graph captures a nuanced vertex-coloring invariant which blends distance constraints with color-class organization, central to applications in frequency assignment, resource allocation, and combinatorial optimization. Formally, the packing chromatic number $\chi_p(G)$ of a graph $G$ is the minimal $k$ such that $V(G)$ can be partitioned into color classes ${V_1, V_2, \dots, V_k}$ with the property that any two vertices within color class $V_i$ are at pairwise distance at least $i+1$. Caterpillars, a fundamental class of trees with path-like topologies and pendant leaves, admit a particularly rich theory, including exact structural characterizations for low packing chromatic number, tight upper bounds, and unresolved algorithmic questions (Furmańczyk et al., 16 Nov 2025).

1. Definitions: Packing Coloring and Distance Constraints

Let $G = (V, E)$ be a finite, simple, undirected graph. For $u, v \in V$, the distance $d_G(u, v)$ is the length of a shortest $(u, v)$-path. The diameter of $G$ is $\max_{u,v} d_G(u,v)$.

An $i$–packing in $G$ is a subset $X \subseteq V$ such that $d_G(x, y) > i$ for all distinct $x, y \in X$. A packing coloring of $G$ with $k$ colors is a mapping $c: V(G) \to \{1, 2, \dots, k\}$ such that, for each $i$, the color class $V_i = \{v \in V(G) : c(v) = i\}$ forms an $i$–packing, i.e., for all $u \ne v$ with $c(u) = c(v) = i$, $d_G(u, v) \ge i+1$. The packing chromatic number $\chi_p(G)$ is the least $k$ for which such a coloring exists.

These constraints generalize classical vertex colorings (the $i=1$ case) and are considerably more restrictive for $i > 1$, particularly in graphs with long induced paths or dense sets of leaves (Furmańczyk et al., 16 Nov 2025).

2. Structure and Notation of Caterpillar Graphs

A caterpillar $C(l; m_1, m_2, \ldots, m_l)$ is defined for integer $l \ge 2$ as follows: begin with a path $P_l = (v_1, v_2, \dots, v_l)$, the backbone, and attach $m_i$ pendant leaves to each backbone vertex $v_i$ ($m_1, m_l \ge 1$, $m_j \ge 0$ for $2 \le j \le l-1$).

The structure is fully determined by the sequence $(m_1, \ldots, m_l)$, requiring at least one pendant at each end to avoid degenerate cases. This model covers all standard caterpillar trees and accommodates arbitrary pendants placement, whose distribution is critical for tight packing colorings.

3. Complete Structural Characterization for $\chi_p(G) \le 3$

The main classification theorem establishes explicit necessary and sufficient conditions for when a caterpillar $G = C(l; m_1, \ldots, m_l)$ satisfies $\chi_p(G) = 3$. The result partitions such caterpillars into seven families, denoted $\mathcal G_1$ through $\mathcal G_7$, defined by the permissible arrangements of the backbone and its pendants. The families depend on backbone length and specified alternations in the sequence $(m_1, \ldots, m_l)$:

• $\mathcal G_1$: $C(4k; m_1, m_2, 0, m_4, 0, m_6, \dots, 0, m_{4k})$, $k \ge 1$
• $\mathcal G_2$: $$C(4k+1; 1, m_2, 0, m_4, \dots, 0, m_{4k}, m_{4k+1})$$, $k \ge 1$
• $\mathcal G_3$: $C(4k+1; m_1, 0, m_3, 0, \dots, 0, m_{4k+1})$, $k \ge 1$
• $\mathcal G_4$: $C(4k+2; m_1, m_2, 0, m_4, \dots, 0, m_{4k+2})$, $k \ge 0$
• $\mathcal G_5$: $$C(4k+3; m_1, m_2, 0, m_4, \dots, 0, m_{4k+2}, m_{4k+3})$$, $k \ge 1$
• $\mathcal G_6$: $C(3; m_1, m_2, 1)$
• $\mathcal G_7$: $C(4k+3; m_1, 0, m_3, 0, \dots, 0, m_{4k+3})$, $k \ge 0$

The proof proceeds by an exhaustive case analysis on starting color pairs assigned to the path ($v_1, v_2$) and shows that extension along the backbone requires a periodic pattern, specifically repetitions of $[1\ 2\ 1\ 3]^*$ or its cyclic variants, with only restricted “end-adjustments.” Attachment of leaves is highly constrained: only backbone vertices colored $2$ or $3$ may receive pendant leaves colored $1$, and at most two leaves can be attached to backbone vertices of color $2$. These structural restrictions yield precisely the seven families, and each admits an explicit 3-packing-coloring constructed via periodic coloring patterns on the backbone, with compatible pendant color assignments (Furmańczyk et al., 16 Nov 2025).

4. Upper Bounds on the Packing Chromatic Number of Caterpillars

Every caterpillar $C$ satisfies the general upper bound $\chi_p(C) \le 7$, established by Sloper, who showed that caterpillars admit an eccentric coloring with at most $7$ colors, a property stronger than packing coloring [Sloper04].

Sharper bounds are available for caterpillars of limited backbone length. For $C(l; m_1, \dots, m_l)$,

• If $l \le 34$, then $\chi_p(C) \le 6$, and there exist examples achieving equality;
• If $l \ge 35$, $\chi_p(C) \le 7$, with this upper bound tight in general.

The method utilizes the packing chromatic number for coronas of paths, $P_l \circ pK_1$, whose exact packing chromatic numbers are known [Laïche et al. 2017], and employs a smoothing argument to show that variable pendant counts per backbone vertex do not surpass this extremal case. Thus, despite the apparent complexity added by arbitrary leaf arrangements, the worst-case packing chromatic number remains bounded and characterized by the proportion of the backbone (Furmańczyk et al., 16 Nov 2025).

5. Open Algorithmic Problems

A central unresolved question is the complexity of recognizing caterpillars with packing chromatic number at most $4$:

• Is there a polynomial-time algorithm to decide, given a caterpillar $G$, whether $\chi_p(G) \le 4$?

For general graphs (and even trees), deciding whether $\chi_p(G) \le k$ is NP-complete for $k \ge 4$ [Fiala & Golovach 2010]. However, the above structural characterization provides a full polynomial-time recognition for $\chi_p(G) \le 3$ in caterpillars via Theorem 5, but it remains unknown if a similar classification and efficient recognition exist for $\chi_p(G) \le 4$.

A plausible implication is that intermediate cases between the trivial polytree and corona-structured extremal caterpillars could hide new worst-case configurations, significant for both theoretical understanding and algorithmic tractability. Determining polynomial-time recognizability for $\chi_p \le 4$ in caterpillars is an open research direction of current interest (Furmańczyk et al., 16 Nov 2025).

6. Historical Context and Related Results

The packing chromatic number was first studied for simple graphs such as paths and cycles, with tight results demonstrating $\chi_p(P_n) \le 3$ and $\chi_p(C_n) \le 4$ [Goddard et al.]. Subsequent work extended analysis to graph products (including path-aligned graph products) and coronas, providing a framework for assessing bounding constants and explicit colorings in more complex families [Laïche et al. 2017]. Sloper's eccentric coloring technique established foundational bounds in trees, directly shaping extremal results for caterpillars. The longstanding interest in the NP-completeness of fixed-threshold recognition informs ongoing focus on efficient algorithms for special tree families, such as caterpillars [Fiala & Golovach 2010; (Furmańczyk et al., 16 Nov 2025)].