แนวคิดหลัก
The paper provides a tight lower bound for the radio number of the Cartesian product of a tree and a complete graph, and gives necessary and sufficient conditions as well as sufficient conditions to achieve this lower bound. The radio number for the Cartesian product of a level-wise regular tree and a complete graph is also determined.
บทคัดย่อ
The paper focuses on the radio labeling problem for the Cartesian product of a tree and a complete graph. The key highlights and insights are:
The paper provides a lower bound for the radio number of the Cartesian product of a tree and a complete graph (Theorem 3.1).
It gives two necessary and sufficient conditions (Theorems 3.2 and 3.3) and three sufficient conditions (Theorem 3.4) to achieve the lower bound.
Using these results, the radio number for the Cartesian product of a level-wise regular tree and a complete graph is determined (Theorem 4.1).
The radio number for the Cartesian product of a path and a complete graph, derived in prior work, can be obtained as a corollary of the results presented in this paper.
The paper introduces key concepts related to radio labeling, such as weight centers of a tree, branches, and level-wise regular trees, and leverages these to derive the main results.
The proofs rely on carefully analyzing the structure of the Cartesian product graph and the properties of the optimal radio labeling.