Yucheng Zhong | Searcharxiv

arXiv2026

Total Vertex Irregularity Strength of Cubic and 4-Regular Graphs

Let $G$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ assigns to each vertex and each edge a label from $\{1,\ldots,k\}$. The weight of a vertex is the sum of its label and the labels of its incident edges. A total labeling is vertex irregular if all vertex weights are distinct. The total vertex irregularity strength $\text{tvs}(G)$ is the smallest $k$ for which $G$ has a vertex irregular total $k$-labeling. For an $r$-regular graph $G$ on $n$ vertices, a counting argument gives $\text{tvs}(G)\ge\lceil(n+r)/(r+1)\rceil$. The restriction of a conjecture of Nurdin, Baskoro, Salman, and Gaos to regular graphs asserts that this bound is attained. We prove this assertion for cubic and $4$-regular graphs. We also show that, for every fixed $r\ge2$, a recent theorem on prescribed degree frequencies implies the assertion for all sufficiently large $r$-regular graphs.

math.CO↗

arXiv2026

Graceful Labeling of Two Families of Spiders

A \emph{graceful labeling} of a graph $G$ is an injective function $f : V(G) \to \{0, \ldots, |E(G)|\}$ such that $\{\,|f(u)-f(v)| : uv \in E(G)\,\} = \{1, \ldots, |E(G)|\}$. If such a labeling exists, then we call $G$ \emph{graceful}. Introduced by Rosa in 1967, graceful labeling has been widely studied, and the Graceful Tree Conjecture asserts that every tree is graceful. The conjecture is known to hold for several classes of trees, including caterpillars, trees with at most four leaves, trees of diameter at most five, and certain spiders. An important subclass is that of \emph{$α$-labelings}, where a graceful labeling $f$ admits an integer $α$ such that each edge joins a vertex with label at most $α$ to one with label greater than $α$. A result from 1982 by Huang, Kotzig, and Rosa shows that if $H$ has an $α$-labeling with a vertex $u$ labeled $0$ or $α$, and $G$ has a graceful labeling with a vertex $v$ labeled $0$, then identifying $u$ and $v$ yields a graceful graph, though this requires a $0$-labeled vertex in $G$. We prove a related result that relaxes this condition: if $G$ has a graceful labeling $f$ such that $f(u)+\lfloor n/2 \rfloor + 1 \le n$ and $n \not\equiv 1 \pmod{4}$, where $u\in V(G)$ and $n\ge 2$ is an integer, then joining $u$ to an end vertex of the vertex-disjoint $n$-vertex path $P_n$ yields a graceful graph. As an application, we show that any spider with legs $L_1,\ldots,L_s$ ($s \ge 1$) satisfying $|E(L_{2})| \ge 2|E(L_1)|+ 4$ and $|E(L_{i+1})| \ge 2|E(L_i)|+ 2$ for $i \in \{2,\ldots, s-1\}$ is graceful. Furthermore, we give an explicit graceful labeling for spiders with one leg of arbitrary length and all others of length at most two such that the center is labeled by $0$. This labeling enables the construction of larger graceful spiders by attaching paths at the center.

math.CO↗