Alain Hertz, Christophe Picouleau
A graceful difference labeling (gdl for short) of a directed graph G with vertex set V is a bijection f between V and {1,...,|V|} such that, when each arc uv is assigned the difference label f(v)-f(u), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.
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We prove via a composition lemma, the Kotzig-Ringel-Rosa conjecture, better known as the Graceful Labeling Conjecture. We also prove via a stronger version of the composition lemma a stronger form of the Graceful Labeling Conjecture.
We prove that if $G$ is a $2r$-regular edge graceful $(p,q)$ graph with $(r,kp)=1$ then $kG$ is edge graceful for odd $k$. We also prove that for certain specific classes of $2r$-regular edge graceful graphs it is possible to drop the requirement that $(r,kp)=1$
In a previous paper the concept of a $d$-divisible graceful $α$-labeling has been introduced as a generalization of classical $α$-labelings and it has been shown how it is useful to obtain certain cyclic graph decompositions. In the present paper it is proved the existence of $d$-divisible graceful $α$-labelings of $C_{4k}\times P_m$ for any integers $k\geq1$, $m\geq2$ for several values of $d$.
In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that {|f(x)-f(y)| | [x,y]\in E(G)} ={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of d=e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call d-graceful α-labeling of a bipartite graph G a d-graceful labeling of G with the property that its maximum value on one of the two bipartite …
We prove the existence of infinite classes of cyclic G-decompositions of the complete multipartite graph, G being a caterpillar, a hairy cycle or a cycle. All the results are obtained by the construction of d-divisible $α$-labelings of G, introduced in [A. Pasotti, On d-graceful labelings, to appear on Ars Combin.] as a generalization of classical $α$-labelings, whose existence implies that one of graph decompositions.
A difference vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy the weight |f(x)-f(y)|. A difference vertex labeling f of a graph G of size n is odd-graceful if f is an injection from V(G) to {0,1,...,2n-1} such that the induced weights are {1,3,...,2n-1}. We show here that any forest whose components are caterpillars is odd-graceful. We also show that every tree of diameter up to five is odd-graceful.
Four algorithms giving rise to graceful graphs from a known (non)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and highly graceful graphs. The RingelRosaKotzig Conjecture is generalized to highly graceful graphs. MayedaSeshu Tree Generation Algorithm is modified to generate all possible graceful labelings of trees of order p. An alternative algorithm in terms of integers modulo p is described which includes all possible graceful labelings of trees of order p and some inte…
In "On the super edge graceful trees of even orders," Chung, Lee, Gao, and Schaffer posed the following problem: Characterize trees of diameter 4 which are super edge-graceful. In this paper, we provide super edge-graceful labelings for all caterpillars and even size lobsters of diameter 4 which permit such labelings. We also provide super edge-graceful labelings for several families of odd size lobsters of diameter 4.
Graceful Labeling of Two Families of Spiders
本文研究了双族蜘蛛的优雅标记问题，提出了一种新的方法，证明了特定结构的蜘蛛是优雅的，并给出了显式的标记方法。
A graph $(p, q)$ graph $G = (V, E)$ is said to be $(k, d)$-hooked Skolem graceful if there exists a bijection $f:V (G)\rightarrow \{1, 2, \dots, p-1, p+1\}$ such that the induced edge labeling $g_f : E \rightarrow \{k, k+d, \dots, k+(n-1)d \}$ defined by $g_f (uv) = |f(u) - f(v)|$ $\forall uv \in E$ is also bijective, where $k$ and $d$ are positive integers. Such a labeling $f$ is called $(k, d)$-hooked Skolem graceful labeling of $G.$ Note that when $k = d = 1$, this notion coincides with that of Hooked Skolem (HS) graceful labeling of the graph G. In this paper, we present some preliminary r…
We settle affirmatively a conjecture posed in [S. M. Hegde, Set colorings of graphs, European Journal of Combinatorics 30 (4) (2009), 986--995]: If some subsets of a set X are assigned injectively to all vertices of a complete bipartite graph G such that the collection of all sets, each of which is the symmetric difference of the sets assigned to the ends of some edge, is the set of all nonempty subsets of X, then G is a star.
One Curious Identity Counting Graceful Labelings
该研究针对素因子含两个不同素数且指数均为n的正整数a、b，证明完全二部图K_{a,b}的本质不同α-优美标号数等于二项式系数四次方的交错和，建立了图论标号与组合恒等式的关联。
Alternating Extremes in Graceful Labelings of Full Binary Trees and Spider Trees
研究完全二叉树和蜘蛛树优美标号中固定形式的交替极值问题，证明了梳状完全二叉树的固定脊柱猜想并计算验证相关完全二叉树，证明蜘蛛树自匹配腿填充定理，给出相关计算结果并表述六臂问题。
On $k$-Super Graceful Labeling of Graphs
该文研究图的$k$-超优美标号问题，给出相关一般性质，得到多类（完全）二部、三部图$k$-超优美性的充分条件，且证明部分条件同时是必要的。
A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. An integer additive set-labeling is an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$, $\mathbb{N}_0$ is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:…
Given a graph $G$, a labeling of $G$ is an injective function $f:V(G)\rightarrow\mathbb{Z}_{\ge 0}$. Under the labeling $f$, the label of a vertex $v$ is $f(v)$, and the induced label of an edge $uv$ is $|f(u) - f(v)|$. The labeling $f$ is graceful if the labels of the vertices are $\{0, 1, \ldots , |V(G)| - 1\}$, and the induced labels of the edges are distinct. The graph $G$ is graceful if it has a graceful labeling. The Graceful Tree Conjecture, introduced by Kotzig in the late 1960's, states that all trees are graceful. It is an open problem whether every diameter-6 tree has a graceful lab…
Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful labeling. In this paper, we study the $k$-super gracefulness of some graphs in which each component is either regular or bi-regular.
Apportionable matrices and gracefully labelled graphs
本文发起对复矩阵分配问题的研究，揭示了其与图的优美标号等图分解问题的联系，证明了秩一矩阵可酉分配，并给出了矩阵可酉分配的必要条件。
We introduce left and right-layered trees as trees with a specific representation and define the excess of a tree. Applying these ideas, we show a range-relaxed graceful labeling which improves on the upper bound for maximum vertex label given by Van Bussel. For the case when the tree is a lobster of size $m$ and diameter $d$, the labeling produces vertex labels no greater than $\frac{3}{2}m-\frac{1}{2}d$. Furthermore, we show that any lobster $T$ with $m$ edges and diameter $d$ has an edge-relaxed graceful bipartite labeling with at least $\max\{\frac{3m-d+6}{4},\frac{5m+d+15}{8}\}$ of the ed…
A graceful n-permutation is a graceful labeling of an n-vertex path P_n. In this paper we improve the asymptotic lower bound on the number of such permutations from (5/3)^n to 2.37^n. This is a computer-assisted proof based on an effective algorithm that enumerates graceful n-permutations. Our algorithm is also presented in detail.
Graceful labelings of spiders with three-edge legs and pendant leaves at the center
本文研究了具有三个边腿和中心悬挂叶的蜘蛛树，证明了此类树的优雅标签性质，并展示了该家族在悬挂叶添加和中心零标签条件下的闭合性。
A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in whi…
Some Results on Additively Graceful Signed Stars and Double Stars
研究有向星和双星的加性优雅标记，针对有向双星提出存在性、唯一性和不存在性结果。
Radio gracefulness of Moore graphs and beyond
该研究围绕无线电频率分配建模的无线电优美标号问题，确定了Moore图等多类低直径高性能计算相关图的无线电优美性，证明了直径3二部图的相关充要条件并算出部分笼图的无线电数。
On the Oberwolfach problem for single-flip $2$-factors via graceful labelings
本文利用优美标号和加倍构造方法，证明了具有单翻转自同构且包含足够大圈的 $2$-正则图的 Oberwolfach 问题及其覆盖与装填版本可解，并显式构造了解。
For $G$ a simple, connected graph, a vertex labeling $f:V(G)\rightarrow \mathbb{Z}_+$ is called a $\textit{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|\geq \operatorname{diam}(G) + 1 - d(u,v)$ for all distinct vertices $u,v\in V(G)$. The $\textit{radio number}$ of $G$ is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,...,|V(G)|\}$ exists, $G$ is called a $\textit{radio graceful graph}$. We determine the radio number of all diameter $3$ Hamming graphs and show that an infinite subset of them is radio graceful.
A proof of the Kotzig-Ringel-Rosa Conjecture
本文通过泛函重构和组合引理，证明了图论中长期存在的Kotzig-Ringel-Rosa猜想，即每棵树都存在优美标号。
In this paper we formulate cycle-supermagic labelings for the disjoint union of isomorphic copies of different families of graphs. We also prove that disjoint union of non isomorphic copies of fans and ladders are cycle-supermagic.
We consider graph labelings with an assignment of odd prime numbers to the vertices. Similarly to graceful graphs, a labeling is said to be elegant if the absolute differences between the labels of adjacent vertices describe exactly the first even numbers. The labels of an elegant tree with $n$ vertices are the first $n$ odd prime numbers and we want that the resulting edge labels are exactly the first even numbers up to $2n-2$. We conjecture that each path is elegant and we give the algorithm with which we got elegant paths of $n$ primes for all $n$ up to $n=3500$.
We answer two questions of Shamik Ghosh in the negative. We show that there exists a lobster tree of diameter less than 6 which accepts no alpha-labeling with two central vertices labeled by the critical number and the maximum vertex label. We also show a simple example of a tree of diameter 4, with an even degree central vertex which does not accept a maximum label in any graceful labeling.
Constructing the antimagic labelings for double stars union paths on three vertices
本文针对反魔法标号问题，确定了使双星图与3顶点路径$P_3$的不交并为反魔法图的$P_3$的确切数量，还给出了顶点和从1到顶点数的$(1,1)$-反魔法标号实例。
Let $G$ be a graph with $m$ edges and let $f$ be a bijection from $E(G)$ to $\{1,2, \dots, m\}$. For any vertex $v$, denote by $ϕ_f(v)$ the sum of $f(e)$ over all edges $e$ incident to $v$. If $ϕ_f(v) \neq ϕ_f(u)$ holds for any two distinct vertices $u$ and $v$, then $f$ is called an {\it antimagic labeling} of $G$. We call $G$ {\it antimagic} if such a labeling exists. Hartsfield and Ringel in 1991 conjectured that all connected graphs except $P_2$ are antimagic. Denote the disjoint union of graphs $G$ and $H$ by $G \cup H$, and the disjoint union of $t$ copies of $G$ by $tG$. For an antimagi…
Stingray Patterns of Dominant Weights
研究固定 e-重量 w 的 sl_r 主要权重集合，证明其分解为由 r 的 compositions 索引的简单形体，并给出闭合计数公式和几何结构。
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