Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2017
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/1708.00624 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910437153439744 |
|---|---|
| author | Lozano, Antoni Mora, Mercè Seara, Carlos |
| author_facet | Lozano, Antoni Mora, Mercè Seara, Carlos |
| contents | A $k$-antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|+k\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. We call a graph $k$-antimagic when it has a $k$-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic, but the conjecture is still open even for trees. Here we study $k$-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use constructive techniques to prove that any caterpillar of order $n$ is $(\lfloor (n-1)/2 \rfloor - 2)$-antimagic. Furthermore, if $C$ is a caterpillar with a spine of order $s$, we prove that when $C$ has at least $\lfloor (3s+1)/2 \rfloor$ leaves or $\lfloor (s-1)/2 \rfloor$ consecutive vertices of degree at most 2 at one end of a longest path, then $C$ is antimagic. As a consequence of a result by Wong and Zhu, we also prove that if $p$ is a prime number, any caterpillar with a spine of order $p$, $p-1$ or $p-2$ is $1$-antimagic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_00624 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Antimagic Labelings of Caterpillars Lozano, Antoni Mora, Mercè Seara, Carlos Combinatorics 68R10 A $k$-antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|+k\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. We call a graph $k$-antimagic when it has a $k$-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic, but the conjecture is still open even for trees. Here we study $k$-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use constructive techniques to prove that any caterpillar of order $n$ is $(\lfloor (n-1)/2 \rfloor - 2)$-antimagic. Furthermore, if $C$ is a caterpillar with a spine of order $s$, we prove that when $C$ has at least $\lfloor (3s+1)/2 \rfloor$ leaves or $\lfloor (s-1)/2 \rfloor$ consecutive vertices of degree at most 2 at one end of a longest path, then $C$ is antimagic. As a consequence of a result by Wong and Zhu, we also prove that if $p$ is a prime number, any caterpillar with a spine of order $p$, $p-1$ or $p-2$ is $1$-antimagic. |
| title | Antimagic Labelings of Caterpillars |
| topic | Combinatorics 68R10 |
| url | https://arxiv.org/abs/1708.00624 |