Antimagic Labelings of Caterpillars

Abstract

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.

Figures (4)

• Figure 1: The number of grey leaves equals the leaves excess $\mathcal{E}(C)$.
• Figure 2: Above, red labels belong to $L_2$; blue labels belong to $L_4$ and green labels belong to $L_1$. For each spine vertex, the partial sum is calculated. Below, labels from $L_3$ (also in green) are assigned to the remaining edges (only the sums are shown) and the final sum of each non-leaf vertex is shown (squared).
• Figure 3: We can derive that $C_1$ is antimagic (from Theorems \ref{['th:caterpillars']} and \ref{['th:LM']}); $C_2$ is antimagic (from Theorem \ref{['th:LM']}), and $2$-antimagic (from Theorem \ref{['th:caterpillars']}); $C_3$ is antimagic (from Theorem \ref{['th:caterpillars']}) and $5$-antimagic (from Theorem \ref{['th:LM']}).
• Figure 4: A caterpillar with a tail of order $8$ (gray vertices). Theorem \ref{['th:tail']} implies that it is antimagic. Nevertheless, we can only derive that it is $2$-antimagic from Theorem \ref{['th:caterpillars']} and $9$-antimagic from Theorem \ref{['th:LM']}. An antimagic labeling is given: red labels correspond to $L_1$, blue labels to $L_3$, and green labels to $L_2$. The vertex sums at vertices belonging to a longest path are shown (squared numbers).

Theorems & Definitions (17)

• Conjecture 1
• Conjecture 2
• Definition 1
• Definition 2
• Theorem \oldthetheorem
• proof
• Corollary 1
• Theorem \oldthetheorem
• Corollary 2
• Corollary 3