[Paper Review] Total Vertex Irregularity Strength of Forests

Marcin Anholcer, Micha L Karo+1 more|arXiv (Cornell University)|Mar 10, 2011

Graph Labeling and Dimension Problems3 references3 citations

TL;DR

This paper establishes the exact total vertex irregularity strength (tvs) for forests without vertices of degree 2 or isolated vertices, proving that tvs(F) = ⌈(n₁ + 1)/2⌉, where n₁ is the number of pendant vertices. The authors construct a labeling using labels 1 to s that ensures all vertices have distinct weighted degrees, leveraging structured edge and vertex labeling strategies based on vertex degree and pendant neighbor counts.

ABSTRACT

We investigate a graph parameter called the total vertex irregularity strength ($tvs(G)$), i.e. the minimal $s$ such that there is a labeling $w: E(G)\cup V(G) ightarrow \{1,2,..,s\}$ of the edges and vertices of $G$ giving distinct weighted degrees $wt_G(v):=w(v)+\sum_{v\in e \in E(G)}w(e)$ for every pair of vertices of $G$. We prove that $tvs(F)=\lceil (n_1+1)/2 ceil$ for every forest $F$ with no vertices of degree 2 and no isolated vertices, where $n_1$ is the number of pendant vertices in $F$. Stronger results for trees were recently proved by Nurdin et al.

Motivation & Objective

• To determine the exact value of the total vertex irregularity strength (tvs) for forests with no vertices of degree 2 and no isolated vertices.
• To extend known bounds for trees to the more general case of forests, including non-tree forests.
• To provide a constructive labeling method that achieves the minimal possible tvs using only labels from {1, 2, ..., s}, where s = ⌈(n₁ + 1)/2⌉.
• To prove that the trivial lower bound on tvs is tight for such forests, thereby closing a gap in prior work.

Proposed method

• Assign label s to all non-pendant edges to standardize high-weight contributions.
• Label pendant edges using a symmetric sequence S = (1, s−1, 2, s−2, ...) to control weighted degree diversity.
• For vertices with j pendant neighbors (j ≥ 1), label j/2 or (j±1)/2 pendant edges with s, depending on parity, to maximize s-label usage without exceeding s total.
• Use greedy vertex labeling to assign weights w(v) such that all total weighted degrees wt(v) = w(v) + sum of incident edge weights are distinct.
• Handle vertices in different degree classes (C_{kj}) separately across weighted degree ranges: [2, 2s], [2s+2, 3s], [3s+1, 4s], and ≥4s+1.
• Leverage degree constraints (e.g., n₁ = 2 + Σ(j−2)n_j for j ≥ 3) to bound vertex counts and ensure sufficient label space for distinct weighted degrees.

Experimental results

Research questions

• RQ1 What is the exact value of the total vertex irregularity strength (tvs) for forests with no vertices of degree 2 and no isolated vertices?
• RQ2 Can the lower bound tvs(F) ≥ ⌈(n₁ + 1)/2⌉ be achieved as the exact value for such forests?
• RQ3 Does the labeling strategy used for trees extend to general forests, and if so, how must it be adapted?
• RQ4 How can vertex weights be assigned greedily across weighted degree ranges to ensure all vertices have distinct total weights?
• RQ5 What structural constraints on vertex degrees (e.g., n₃₀, n₄₃, n₄₄) ensure sufficient label space for distinct weighted degrees?

Key findings

• The total vertex irregularity strength of any forest F with no vertices of degree 2 and no isolated vertices is exactly tvs(F) = ⌈(n₁ + 1)/2⌉.
• The lower bound tvs(F) ≥ ⌈(n₁ + 1)/2⌉ is tight and achievable via a constructive labeling method.
• The labeling strategy uses only labels from {1, 2, ..., s} with s = ⌈(n₁ + 1)/2⌉, and ensures all vertices have distinct weighted degrees.
• Vertices in C_{1} (pendant vertices) have weighted degrees in [2, 2s], while all others have weighted degrees ≥ 2s + 2.
• The method guarantees sufficient label space in each weighted degree range (e.g., [3s+1, 4s]) by bounding vertex counts using degree constraints.
• The result extends to forests with at most one vertex of degree 2, as shown in the generalized Theorem 1, where tvs(F) remains ⌈(n₁ + 1)/2⌉.