Judicious Partitions in Edge-Weighted Graphs with Bounded Maximum Weighted Degree
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arXiv
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| Format: | Preprint |
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2025
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| author | Gutin, G. Nielsen, M. A. Yeo, A. Zhou, Y. |
| author_facet | Gutin, G. Nielsen, M. A. Yeo, A. Zhou, Y. |
| contents | In this paper, we investigate bounds for the following judicious $k$-partitioning problem: Given an edge-weighted graph $G$, find a $k$-partition $(V_1,V_2,\dots ,V_k)$ of $V(G)$ such that the total weight of edges in the heaviest induced subgraph, $\max_{i=1}^k w(G[V_i])$, is minimized. In our bounds, we also take into account the weight $w(V_1,V_2,\dots,V_k)$ of the cut induced by the partition (i.e., the total weight of edges with endpoints in different parts) and show the existence of a partition satisfying tight bounds for both quantities simultaneously. We establish such tight bounds for the case $k=2$ and, to the best of our knowledge, present the first (even for unweighted graphs) completely tight bound for $k=3$. We also show that, in general, these results cannot be extended to $k \geq 4$ without introducing an additional lower-order term, and we propose a corresponding conjecture. Moreover, we prove that there always exists a $k$-partition satisfying $\max \left\{ w(G[V_i]) : i \in [k] \right\} \leq \frac{w(G)}{k^2} + \frac{k - 1}{2k^2} Δ_w(G),$ where $Δ_w(G)$ denotes the maximum weighted degree of $G$. This bound is tight for every integer $k\geq 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_05827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Judicious Partitions in Edge-Weighted Graphs with Bounded Maximum Weighted Degree Gutin, G. Nielsen, M. A. Yeo, A. Zhou, Y. Combinatorics In this paper, we investigate bounds for the following judicious $k$-partitioning problem: Given an edge-weighted graph $G$, find a $k$-partition $(V_1,V_2,\dots ,V_k)$ of $V(G)$ such that the total weight of edges in the heaviest induced subgraph, $\max_{i=1}^k w(G[V_i])$, is minimized. In our bounds, we also take into account the weight $w(V_1,V_2,\dots,V_k)$ of the cut induced by the partition (i.e., the total weight of edges with endpoints in different parts) and show the existence of a partition satisfying tight bounds for both quantities simultaneously. We establish such tight bounds for the case $k=2$ and, to the best of our knowledge, present the first (even for unweighted graphs) completely tight bound for $k=3$. We also show that, in general, these results cannot be extended to $k \geq 4$ without introducing an additional lower-order term, and we propose a corresponding conjecture. Moreover, we prove that there always exists a $k$-partition satisfying $\max \left\{ w(G[V_i]) : i \in [k] \right\} \leq \frac{w(G)}{k^2} + \frac{k - 1}{2k^2} Δ_w(G),$ where $Δ_w(G)$ denotes the maximum weighted degree of $G$. This bound is tight for every integer $k\geq 2$. |
| title | Judicious Partitions in Edge-Weighted Graphs with Bounded Maximum Weighted Degree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.05827 |