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1. Hyperbolic families and coloring graphs on surfaces

   https://doi.org/https://doi.org/10.1090/btran/26  

   Postle, Luke; Thomas, Robin (December 2018, Transactions of the American Mathematical Society. Series B)

   We develop a theory of linear isoperimetric inequalities for graphs on surfaces and apply it to coloring problems, as follows. Let $ G$ be a graph embedded in a fixed surface $$ \Sigma $$ of genus $ g$ and let $$ L=(L(v):v\in V(G))$$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $ G$ is triangle-free, or each list has size at least three and $ G$ has no cycle of length four or less. An $ L$-coloring of $ G$ is a mapping $$ \phi $$ with domain $ V(G)$ such that $$ \phi (v)\in L(v)$$ for every $$ v\in V(G)$$ and $$ \phi (v)\ne \phi (u)$$ for every pair of adjacent vertices $$ u,v\in V(G)$$. We prove if every non-null-homotopic cycle in $ G$ has length $$ \Omega (\log g)$$, then $ G$ has an $ L$-coloring, if $ G$ does not have an $ L$-coloring, but every proper subgraph does (``$ L$-critical graph''), then $$ \vert V(G)\vert=O(g)$$, if every non-null-homotopic cycle in $ G$ has length $$ \Omega (g)$$, and a set $$ X\subseteq V(G)$$ of vertices that are pairwise at distance $$ \Omega (1)$$ is precolored from the corresponding lists, then the precoloring extends to an $ L$-coloring of $ G$, if every non-null-homotopic cycle in $ G$ has length $$ \Omega (g)$$, and the graph $ G$ is allowed to have crossings, but every two crossings are at distance $$ \Omega (1)$$, then $ G$ has an $ L$-coloring, if $ G$ has at least one $ L$-coloring, then it has at least $$ 2^{\Omega (\vert V(G)\vert)}$$ distinct $ L$-colorings. We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $ L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities. 

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2. Pure Pairs. V. Excluding Some Long Subdivision

   https://doi.org/10.1007/s00493-023-00025-8  

   Scott, Alex; Seymour, Paul; Spirkl, Sophie (June 2023, Combinatorica)

   A “pure pair” in a graph G is a pair A, B of disjoint subsets of V(G) such that A is complete or anticomplete to B. Jacob Fox showed that for all ε>0, there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A, B with |A|,|B|≥εn. He also proved that for all c>0 there exists ε>0 such that for every comparability graph G with n>1 vertices, there is a pure pair A, B with |A|,|B|≥εn1−c; and conjectured that the same holds for every perfect graph G. We prove this conjecture and strengthen it in several ways. In particular, we show that for all c>0, and all ℓ1,ℓ2≥4/c+9, there exists ε>0 such that, if G is an (n>1)-vertex graph with no hole of length exactly ℓ1 and no antihole of length exactly ℓ2, then there is a pure pair A, B in G with |A|≥εn and |B|≥εn1−c. This is further strengthened, replacing excluding a hole by excluding some “long” subdivision of a general graph. 

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3. A Spanner for the Day After

   https://doi.org/10.1007/s00454-020-00228-6  

   Buchin, Kevin; Har-Peled, Sariel; Oláh, Dániel (December 2020, Discrete & Computational Geometry)

   null (Ed.)

   Abstract We show how to construct a $$(1+\varepsilon )$$ ( 1 + ε ) -spanner over a set $${P}$$ P of n points in $${\mathbb {R}}^d$$ R d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters $${\vartheta },\varepsilon \in (0,1)$$ ϑ , ε ∈ ( 0 , 1 ) , the computed spanner $${G}$$ G has $$\begin{aligned} {{\mathcal {O}}}\bigl (\varepsilon ^{-O(d)} {\vartheta }^{-6} n(\log \log n)^6 \log n \bigr ) \end{aligned}$$ O ( ε - O ( d ) ϑ - 6 n ( log log n ) 6 log n ) edges. Furthermore, for any k , and any deleted set $${{B}}\subseteq {P}$$ B ⊆ P of k points, the residual graph $${G}\setminus {{B}}$$ G \ B is a $$(1+\varepsilon )$$ ( 1 + ε ) -spanner for all the points of $${P}$$ P except for $$(1+{\vartheta })k$$ ( 1 + ϑ ) k of them. No previous constructions, beyond the trivial clique with $${{\mathcal {O}}}(n^2)$$ O ( n 2 ) edges, were known with this resilience property (i.e., only a tiny additional fraction of vertices, $$\vartheta |B|$$ ϑ | B | , lose their distance preserving connectivity). Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black-box fashion. 

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4. Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

   https://doi.org/10.1017/S0963548320000061  

   Falgas-Ravry, Victor; Markström, Klas; Zhao, Yi (March 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F , what is c 1 ( n , F ), the least integer d such that if G is an n -vertex 3-graph with minimum vertex-degree $$\delta_1(G)>d$$ then every vertex of G is contained in a copy of F in G ? We asymptotically determine c 1 ( n , F ) when F is the generalized triangle $$K_4^{(3)-}$$ , and we give close to optimal bounds in the case where F is the tetrahedron $$K_4^{(3)}$$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n -vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case. 

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5. A Simple Deterministic Near-Linear Time Approximation Scheme for Transshipment with Arbitrary Positive Edge Costs

   https://doi.org/10.4230/LIPIcs.ESA.2024.56  

   Fox, Emily (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   We describe a simple deterministic near-linear time approximation scheme for uncapacitated minimum cost flow in undirected graphs with positive real edge weights, a problem also known as transshipment. Specifically, our algorithm takes as input a (connected) undirected graph G = (V, E), vertex demands b ∈ R^V such that ∑_{v ∈ V} b(v) = 0, positive edge costs c ∈ R_{≥ 0}^E, and a parameter ε > 0. In O(ε^{-2} m log^{O(1)} n) time, it returns a flow f such that the net flow out of each vertex is equal to the vertex’s demand and the cost of the flow is within a (1 ± ε) factor of optimal. Our algorithm is combinatorial and has no running time dependency on the demands or edge costs. With the exception of a recent result presented at STOC 2022 for polynomially bounded edge weights, all almost- and near-linear time approximation schemes for transshipment relied on randomization to embed the problem instance into low-dimensional space. Our algorithm instead deterministically approximates the cost of routing decisions that would be made if the input were subject to a random tree embedding. To avoid computing the Ω(n²) vertex-vertex distances that an approximation of this kind suggests, we also take advantage of the clustering method used in the well-known Thorup-Zwick distance oracle. 

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