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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. Large monochromatic components in hypergraphs with large minimum codegree

   https://doi.org/10.1002/jgt.23044  

   Bal, Deepak; DeBiasio, Louis (October 2023, Journal of Graph Theory)

   Abstract A result of Gyárfás says that for every 3‐coloring of the edges of the complete graph , there is a monochromatic component of order at least , and this is best possible when 4 divides . Furthermore, for all and every ‐coloring of the edges of the complete ‐uniform hypergraph , there is a monochromatic component of order at least and this is best possible for all . Recently, Guggiari and Scott and independently Rahimi proved a strengthening of the graph case in the result above which says that the same conclusion holds if is replaced by any graph on vertices with minimum degree at least ; furthermore, this bound on the minimum degree is best possible. We prove a strengthening of the case in the result above which says that the same conclusion holds if is replaced by any ‐uniform hypergraph on vertices with minimum ‐degree at least ; furthermore, this bound on the ‐degree is best possible. 

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3. Complexity of Ck-Coloring in Hereditary Classes of Graphs

   Chudnovsky, M.; Huang, S.; Rzazewski, P.; Spirkl, S.; and Zhong, M. (September 2019, European Symposium on Algorithms)

   For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f : V (G) → V (H) such that for every edge uv ∈ E(G) it holds that f(u)f(v) ∈ E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of Pt-free graphs. We show that for every odd k ≥ 5 the Ck-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P9-free graphs. On the other hand, we prove that the extension version of Ck-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw. 

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4. On Weak Flexibility in Planar Graphs

   https://doi.org/10.1007/s00373-022-02564-1  

   Lidický, Bernard; Masařík, Tomáš; Murphy, Kyle; Zerbib, Shira (December 2022, Graphs and Combinatorics)

   Abstract Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs (J Graph Theory 92(3):191–206, 2019, https://doi.org/10.1002/jgt.22447 ). In this new setting, each vertex v in some subset of V ( G ) has a request for a certain color r ( v ) in its list of colors L ( v ). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal constant $$\varepsilon >0$$ ε > 0 such that any graph G in some graph class $$\mathscr {C}$$ C satisfies at least $$\varepsilon$$ ε proportion of the requests. More formally, for $$k > 0$$ k > 0 the goal is to prove that for any graph $$G \in \mathscr {C}$$ G ∈ C on vertex set V , with any list assignment L of size k for each vertex, and for every $$R \subseteq V$$ R ⊆ V and a request vector $$(r(v): v\in R, ~r(v) \in L(v))$$ ( r ( v ) : v ∈ R , r ( v ) ∈ L ( v ) ) , there exists an L -coloring of G satisfying at least $$\varepsilon |R|$$ ε | R | requests. If this is true, then $$\mathscr {C}$$ C is called $$\varepsilon$$ ε - flexible for lists of size k . Choi, Clemen, Ferrara, Horn, Ma, and Masařík (Discrete Appl Math 306:20–132, 2022, https://doi.org/10.1016/j.dam.2021.09.021 ) introduced the notion of weak flexibility , where $$R = V$$ R = V . We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists $$\varepsilon (b)>0$$ ε ( b ) > 0 so that the class of planar graphs without $$K_4, C_5 , C_6 , C_7, B_b$$ K 4 , C 5 , C 6 , C 7 , B b is weakly $$\varepsilon (b)$$ ε ( b ) -flexible for lists of size 4 (here $$K_n$$ K n , $$C_n$$ C n and $$B_n$$ B n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without $$K_4, C_5 , C_6 , C_7, B_5$$ K 4 , C 5 , C 6 , C 7 , B 5 is $$\varepsilon$$ ε -flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable. 

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5. Monochromatic Subgraphs in Iterated Triangulations

   https://doi.org/10.37236/9292  

   Ma, Jie; Tang, Tianyun; Yu, Xingxing (October 2020, The Electronic Journal of Combinatorics)

   null (Ed.)

   For integers $$n\ge 0$$, an iterated triangulation $$\mathrm{Tr}(n)$$ is defined recursively as follows: $$\mathrm{Tr}(0)$$ is the plane triangulation on three vertices and, for $$n\ge 1$$, $$\mathrm{Tr}(n)$$ is the plane triangulation obtained from the plane triangulation $$\mathrm{Tr}(n-1)$$ by, for each inner face $$F$$ of $$\mathrm{Tr}(n-1)$$, adding inside $$F$$ a new vertex and three edges joining this new vertex to the three vertices incident with $$F$$. In this paper, we show that there exists a 2-edge-coloring of $$\mathrm{Tr}(n)$$ such that $$\mathrm{Tr}(n)$$ contains no monochromatic copy of the cycle $$C_k$$ for any $$k\ge 5$$. As a consequence, the answer to one of two questions asked by Axenovich et al. is negative. We also determine the radius 2 graphs $$H$$ for which there exists $$n$$ such that every 2-edge-coloring of $$\mathrm{Tr}(n)$$ contains a monochromatic copy of $$H$$, extending a result of Axenovich et al. for radius 2 trees. 

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