An Inductive Approach to Strongly Antimagic Labelings of Graphs

An antimagic labeling for a graph $G$ with $m$ edges is a bijection $f: E(G) \to \{1, 2, \dots, m\}$ so that $\phi_f(u) \neq \phi_f(v)$ holds for any pair of distinct vertices $u, v \in V(G)$, where $\phi_f(x) = \sum_{x \in e} f(e)$. A strongly antimagic labeling is an antimagic labeling with an additional condition: For any $u, v \in V(G)$, if $\deg(u) > \deg(v)$, then $\phi_f(u) > \phi_f(v)$. A graph $G$ is strongly antimagic if it admits a strongly antimagic labeling. We present inductive properties of strongly antimagic labelings of graphs. This approach leads to simplified proofs that spiders and double spiders are strongly antimagic, previously shown by Shang [Spiders are antimagic, Ars Combinatoria, 118 (2015), 367--372] and Huang [Antimagic labeling on spiders, Master's Thesis, Department of Mathematics, National Taiwan University, 2015], and by Chang, Chin, Li and Pan [The strongly antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68], respectively. We fix a subtle error in [The strongly antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68]. Further, we prove certain level-wise regular trees, cycle spiders and cycle double spiders are all strongly antimagic

Antimagic Labeling of Forests

An antimagic labeling of a graph G(V,E) is a bijection f mapping from E to the set {1,2,…, |E|}, so that for any two different vertices u and v, the sum of f(e) over all edges e incident to u, and the sum of f(e) over all edges e incident to v, are distinct.  We call G antimagic if it admits an antimagic labeling. A forest is a graph without cycles; equivalently, every component of a forest is a tree. It was proved by Kaplan, Lev, and Roditty in 2009, and by Liang, Wong, and Zhu in 2014 that every tree with at most one vertex of degree two is antimagic. A major tool used in the proof is the zero-sum partition introduced by Kaplan, Lev, and Roditty in 2009. In this article, we provide an algorithmic representation for the zero-sum partition method and apply this method to show that every forest with at most one vertex of degree two is also antimagic

On (d,1)-total numbers of graphs

AbstractA (d,1)-total labelling of a graph G assigns integers to the vertices and edges of G such that adjacent vertices receive distinct labels, adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least d. The span of a (d,1)-total labelling is the maximum difference between two labels. The (d,1)-total number, denoted λdT(G), is defined to be the least span among all (d,1)-total labellings of G. We prove new upper bounds for λdT(G), compute some λdT(Km,n) for complete bipartite graphs Km,n, and completely determine all λdT(Km,n) for d=1,2,3. We also propose a conjecture on an upper bound for λdT(G) in terms of the chromatic number and the chromatic index of G

Antimagic Labeling for Unions of Graphs with Many Three-Paths

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 $\phi_f(v)$ the sum of $f(e)$ over all edges $e$ incident to $v$. If $\phi_f(v) \neq \phi_f(u)$ 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 antimagic graph $G$ (connected or disconnected), we define the parameter $\tau(G)$ to be the maximum integer such that $G \cup tP_3$ is antimagic for all $t \leq \tau(G)$. Chang, Chen, Li, and Pan showed that for all antimagic graphs $G$, $\tau(G)$ is finite [Graphs and Combinatorics 37 (2021), 1065--1182]. Further, Shang, Lin, Liaw [Util. Math. 97 (2015), 373--385] and Li [Master Thesis, National Chung Hsing University, Taiwan, 2019] found the exact value of $\tau(G)$ for special families of graphs: star forests and balanced double stars respectively. They did this by finding explicit antimagic labelings of $G\cup tP_3$ and proving a tight upper bound on $\tau(G)$ for these special families. In the present paper, we generalize their results by proving an upper bound on $\tau(G)$ for all graphs. For star forests and balanced double stars, this general bound is equivalent to the bounds given in \cite{star forest} and \cite{double star} and tight. In addition, we prove that the general bound is also tight for every other graph we have studied, including an infinite family of jellyfish graphs, cycles $C_n$ where $3 \leq n \leq 9$, and the double triangle $2C_3$

Circular chromatic number for iterated Mycielski graphs

For a graph G, let M(G) denote the Mycielski graph of G. The t-th iterated Mycielski graph of G, M t (G), is defined recursively by M 0 (G) = G, and M t (G)= M(M t−1 (G)) for t ≥ 1. Let χc(G) denote the circular chromatic number of G. We prove two main results: 1) If G has a universal vertex x, then χc(M(G)) = χ(M(G)) if χc(G − x)> χ(G) − 1/2 and G is not a star, otherwise χc(M(G)) = χ(M(G)) − 1/2; and 2) χc(M t (Km)) = χ(M t (Km)) if m ≥ 2 t−1 + 2t − 2 and t ≥ 2

T-colorings of graphs

AbstractGiven a finite set T of positive integers containing {0};, a T-coloring of a simple graph G is a nonnegative integer function f defined on the vertex set of G, such that if {u, v}; ϵ E(G) then ¦f(u) - f(v)¦ ∉ T. The T-span of a T-coloring is defined as the difference of the largest and smallest colors used; the T-span of G, spT(G), is the minimum span over all T-colorings of G. It is known that the T-span of G satisfies spT(Kω(G)) ⩽ spT(G) ⩽ spT(Kx(G)). When T is an r-initial set (Cozzens and Roberts, 1982), or a k multiple of s set (A. Raychaudhuri, 1985), then spT(G) = spT(Kx(G)) for all graphsG. Using graph homomorphisms and a special family of graphs, we characterize those T's with equality spT(G) =spT(Kx(G)) for all graphs G. We discover new T's with the same result. Furthermore, we get a necessary and sufficient condition of equality spT(G) = spT(Km) for all graphsG with X(G) = m

Hamiltonian Spectra of Trees

Let G be a connected graph, and let d(u,v) denote the distance between vertices u and v in G. For any cyclic ordering π of V (G), π = (v1,v2, · · ·,vn,vn+1) where vn+1 = v1, let d(π) = n� d(vi,vi+1). The set of possible values of d(π) over all cyclic orderings π of V (G) is called the Hamiltonian spectrum of G. We determine the Hamiltonian spectrum for any tree.

Sizes of graphs with fixed ordered and spans for circular-distance-two labelings

A k-circular-distance-two labeling (or k-c-labeling) of a simple graph G is a vertex-labeling, using the labels 0, 1, 2, · · · , k − 1, such that the “circular difference” (mod k) of the labels for adjacent vertices is at least two, and for vertices of distance-two apart is at least one. The σ-number, σ(G), of a graph G is the minimum k of a k-c-labeling of G. For any given positive integers n and k, let G σ (n, k) denote the set of graphs G on n vertices and σ(G) = k. We determine the maximum size (number of edges) and the minimum size of a graph G ∈ G σ (n, k). Furthermore, we prove that for any value p between the maximum and the minimum size, there exists a graph G ∈ G σ (n, k) of size p. These results are analogues of the ones by Georges and Mauro [4] on distance-two labelings. Keywords. Vertex-labeling, circular difference, circular-distance-two labeling, distance-two labeling.

Fractional chromatic number of distance graphs generated by two-interval sets

AbstractLet D be a set of positive integers. The distance graph generated by D, denoted by G(Z,D), has the set Z of all integers as the vertex set, and two vertices x and y are adjacent whenever |x−y|∈D. For integers 1<a≤b<m−1, define Da,b,m={1,2,…,a−1}∪{b+1,b+2,…,m−1}. For the special case a=b, the chromatic number for the family of distance graphs G(Z,Da,a,m) was first studied by R.B. Eggleton, P. Erdős and D.K. Skilton [Colouring the real line, J. Combin. Theory (B) 39 (1985) 86–100] and was completely solved by G. Chang, D. Liu and X. Zhu [Distance graphs and T-coloring, J. Combin. Theory (B) 75 (1999) 159–169]. For the general case a≤b, the fractional chromatic number for G(Z,Da,b,m) was studied by P. Lam and W. Lin [Coloring distance graphs with intervals as distance sets, European J. Combin. 26 (2005) 25 1216–1229] and by J. Wu and W. Lin [Circular chromatic numbers and fractional chromatic numbers of distance graphs with distance sets missing an interval, Ars Combin. 70 (2004) 161–168], in which partial results for special values of a,b,m were obtained. In this article, we completely settle this problem for all possible values of a,b,m