← 研踪 PaperTrailarXiv:2610.25863 · v2
On Multi-label radio Labelings of Graphs
首次提交 2026-10-19 14:18
版本更新 2026-11-06 17:29
雷达观测 2026-11-09 11:19
摘要
We propose and study a multi-labeling generalization of the $k$-radio labeling problem of graphs: given a graph $G$ and integers $k$ and $m$, assign to each vertex of $G$ a set of $m$ integer labels such that distinct labels assigned to a vertex differ by at least $k+1$, and the labels of any two distinct vertices $x$ and $y$ pairwise differ by at least $k+1-d(x,y)$, with $d(x,y)$ being the distance between vertices $x$ and $y$. We prove general bounds on the minimum span of such a labeling and more refined results for the case $k=1$. For paths and cycles, we derive sharp bounds on the minimum span when $k$ is small. We finally propose a conjecture for the minimum span on the path and for an almost-tight bound on the cycle.
作者备注：14 pages
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v2 · 2026-11-06 17:29 · 最新观测版本
On Multi-label radio Labelings of Graphs
We propose and study a multi-labeling generalization of the $k$-radio labeling problem of graphs: given a graph $G$ and integers $k$ and $m$, assign to each vertex of $G$ a set of $m$ integer labels such that distinct labels assigned to a vertex differ by at least $k+1$, and the labels of any two distinct vertices $x$ and $y$ pairwise differ by at least $k+1-d(x,y)$, with $d(x,y)$ being the distance between vertices $x$ and $y$. We prove general bounds on the minimum span of such a labeling and more refined results for the case $k=1$. For paths and cycles, we derive sharp bounds on the minimum span when $k$ is small. We finally propose a conjecture for the minimum span on the path and for an almost-tight bound on the cycle.
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