On trees with a maximum proper partial 0-1 coloring containing a maximum matching

I prove that in a tree in which the distance between any two endpoints is even, there is a maximum proper partial 0-1 coloring such that the edges colored by 0 form a maximum matching.

💡 Research Summary

The paper introduces a novel concept called a partial 0‑1 coloring of a graph. Given a graph G, a mapping f from a subset X of its edges to the set {0,1} is called a partial 0‑1 coloring. The sets f⁰ = {e ∈ X | f(e)=0} and f¹ = {e ∈ X | f(e)=1} are required to be matchings; when this holds the coloring is said to be proper. Among all proper partial 0‑1 colorings, those that maximize the total number of colored edges |f⁰∪f¹| are called maximum proper partial (MPP) colorings. For an MPP coloring f we denote |f⁰| by Ψ(G) and |f¹| by Δ(G). It is immediate that Ψ(G) ≤ ν(G), where ν(G) is the size of a maximum matching of G, but equality does not hold for arbitrary graphs.

The main result of the paper is that for a special class of trees—those in which the distance between any two leaves (endpoints) is an even number—the equality Ψ(G)=ν(G) always holds. In other words, there exists an MPP coloring in which the edges colored 0 form a maximum matching.

The proof proceeds through a series of lemmas that allow one to manipulate an existing MPP coloring so that certain prescribed edges become colored 0. Lemma 1 shows that if a vertex u has degree 1 and is adjacent to w, then there is an MPP coloring with (u,w) ∈ f⁰. The argument uses a standard alternating‑path exchange: if (u,w) is not already in f⁰, either it lies in f¹ (in which case a neighboring edge of w belonging to f⁰ is swapped) or it is uncolored, and a maximal alternating path starting at w is flipped. Lemma 2 treats the configuration where two pendant vertices u and v share a common neighbor w. It proves that one can force (u,w)∈f⁰ and (v,w)∈f¹ while increasing both Ψ and Δ by one, and it also establishes the recurrence relations Ψ(G)=1+Ψ(G{u,v,w}) and Δ(G)=2+Δ(G{u,v,w}). From these lemmas a corollary is derived for a path of length four (vertices u₀,…,u₄): Ψ(G)=Ψ(G\U)+4 and Ψ(G)≥2+Ψ(G\U).

The central theorem is proved by induction on the number of edges |E(G)|. The base case |E(G)|≤6 is verified directly. For the inductive step, assume the statement holds for all trees with fewer than t edges (t≥7) and consider a tree G with |E(G)|=t satisfying the even‑leaf‑distance condition. Six exhaustive structural cases are examined, each identifying a specific sub‑configuration U (ranging from a simple three‑vertex path to more intricate ten‑vertex patterns). In each case the tree is reduced to a smaller tree G′ by deleting U or a few edges. The induction hypothesis guarantees Ψ(G′)=ν(G′). Using Lemma 1, Lemma 2, and the corollary, the coloring of G′ is extended to a coloring of G that preserves the maximality of |f⁰|. Consequently Ψ(G)≥ν(G). Since the opposite inequality Ψ(G)≤ν(G) holds for any graph, equality follows.

The even‑distance condition is crucial: it forces the pendant structures to appear in pairs that can be matched by alternating‑path exchanges, ensuring that the 0‑colored edges can always be arranged into a maximum matching. The proof technique mirrors classic matching theory (alternating paths, augmenting paths) but is recast in the language of 0‑1 edge colorings, offering a fresh perspective.

Overall, the paper establishes a clear structural link between maximum proper partial 0‑1 colorings and maximum matchings for a non‑trivial class of trees. It enriches the theory of graph colorings by showing that, under suitable parity constraints, the seemingly weaker requirement of a proper partial 0‑1 coloring already forces the 0‑colored edges to be optimal with respect to matching size. This result opens avenues for further research, such as relaxing the parity condition, extending the analysis to general graphs, or exploring partial colorings with more than two colors.