Caterpillars are Antimagic

Lozano, Antoni; Mora, Mercè; Seara, Carlos; Tey, Joaquín

doi:10.1007/s00009-020-01688-z

Cited by 15 publications

(11 citation statements)

References 24 publications

(22 reference statements)

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“…This was followed by Bérczi et al [BBV15] and Chang et al [CLPZ16] proving that k-regular graphs are antimagic, when k is even and k ≥ 4. Following partial results of Deng and Li [DL19] and Lozano et al [LMS19], Lozano et al [LMST21] proved that all caterpillars are antimagic.…”

Section: Introductionmentioning

confidence: 91%

List-antimagic labeling of vertex-weighted graphs

Berikkyzy¹,

Brandt²,

Jahanbekam³

et al. 2021

Discrete Mathematics &Amp; Theoretical Computer Science

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A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.

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Section: Introductionmentioning

confidence: 91%

List-antimagic labeling of vertex-weighted graphs

Berikkyzy¹,

Brandt²,

Jahanbekam³

et al. 2021

Discrete Mathematics &Amp; Theoretical Computer Science

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“…Step 2: Label the edges in the legs except one incident with each vertex in U (6) for all v ∈ U do (7) for all legs e incident with v except one do (8) a random label from ([1] ∪ [⌈ℓ/2⌉ + 2, m − ⌊ℓ/2⌋])\φ(E(T)) ⟶ φ(e)…”

Section: An Algorithmmentioning

confidence: 99%

“…is topic was investigated by many researchers; for instance, see [2][3][4][5][6]. Recently, Lozano et al [7] proved that caterpillars are antimagic, where a caterpillar is a tree with at least three vertices such that the removal of its leaves produces a path.…”

Section: Introductionmentioning

confidence: 99%

Product Antimagic Labeling of Caterpillars

Wang

Gao

2021

Journal of Mathematics

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Let G be a graph with m edges. A product antimagic labeling of G is a bijection from the edge set E G to the set 1,2 , … , m such that the vertex-products are pairwise distinct, where the vertex-product of a vertex v is the product of labels on the incident edges of v . A graph is called product antimagic if it admits a product antimagic labeling. In this paper, we will show that caterpillars with at least three edges are product antimagic by an O m log m algorithm.

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“…On the other hand, Liang, Wong, and Zhu [9] proved that a tree with many vertices of degree 2 is antimagic. The latest result on antimagic trees is by Lozano, Mora, Seara, and Tey [10] who proved that caterpillars are antimagic.…”

Section: Introductionmentioning

confidence: 99%

Another Antimagic Conjecture

et al. 2021

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An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

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Caterpillars are Antimagic

Cited by 15 publications

References 24 publications

List-antimagic labeling of vertex-weighted graphs

List-antimagic labeling of vertex-weighted graphs

Product Antimagic Labeling of Caterpillars

Another Antimagic Conjecture

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