Strongly connected orientation with minimum lexicographic order of indegrees

Egalitarian Graph Orientations

Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order o…

Proper Orientations of Planar Bipartite Graphs

An orientation of a graph $G$ is proper if any two adjacent vertices have different indegrees. The proper orientation number $\overrightarrowχ(G)$ of a graph $G$ is the minimum of the maximum indegree, taken over all proper orientations of $G$. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientatio…

入度多面体上的可分离凸优化

Separable convex optimization over indegree polytopes

本文研究无向图在入度目标下的无环定向优化，证明多种公平性目标（如凸函数和、递减最小化等）的NP难度，同时给出加权最小最后排序等多项式算法及近似算法，刻画了算法边界并应用于调度与无死锁路由。

Beyond Degree Choosability

Let $G$ be a connected graph with maximum degree $Δ$. Brooks' theorem states that $G$ has a $Δ$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is \emph{degree-choosable} if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a \emph{Gallai tree}. This degree-choosability result was further strengthened to Alon--Tars…

Seymour紧导向

Seymour-tight orientations

研究Seymour第二邻域猜想的几乎反例，探讨Seymour紧导向的性质及其在勒克西克图形积中的封闭性，并构造反例。