A family of paths strongly separates the edges of a graph if every two distinct edges are separated in both directions by paths in the family. Bonamy, Botler, Dross, Naia, and Skokan proved that every $n$-vertex graph admits such a family of at most $19n$ paths. We improve this bound to $10n-o(n)$.
A forest in an edge-colored graph is rainbow if its edges have pairwise distinct colors. We prove that for every $\varepsilon>0$ and all sufficiently large integers $m$, every properly edge-colored simple graph with $km$ edges, where $1\leq k\leq 2m$ and every color class has size at most $m$, contains at least $(1-\va...
A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all...
Let $k\geq2$ be an integer. A $1\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $\chi'_k(G)$ denote the minimum number of colors required, and let $\chi'_k$ be the supremum of $\chi'_k(G)$ over all finite simple graphs $G$....
A rigorous Lipschitz analysis for constant curvature graph convolutional networks ( \(\kappa\) -GCNs) and enhances their robustness through Lipschitz regularization and derives upper bounds for the Lipschitz constants across constant curvature spaces, thereby standardizing the Lipschitz limits of the \(\kappa\) -stereo...
Yang Shi, Jingchao Wang, Liangsi Lu et al.· ACM Transactions on Knowledg...· 0 citations
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