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1. Three coloring via triangle counting

   Hamaker, Zachary; Vatter, Vincent (August 2023, The Australasian journal of combinatorics)

   Albert, Michael; Billington, Elizabeth J (Ed.)

   In the first partial result toward Steinberg’s now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly fewer than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey’s resolution of Ore’s conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable. 

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2. 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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3. Graph coloring and semidefinite rank

   https://doi.org/10.1007/s10107-024-02085-0  

   Mirka, Renee; Smedira, Devin; Williamson, David P (July 2024, Mathematical Programming)

   This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger, Motwani, and Sudan give a vector program in which a coloring of a graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have rank high enough that the primal solution encodes a coloring. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a $$k$$-tree has sufficiently high dual rank, and we can extract the coloring from the corresponding low-rank primal solution. We can also show that if a graph is not uniquely colorable, then no sufficiently high rank dual optimal solution can exist. This allows us to completely characterize the planar graphs for which dual optimal solutions have sufficiently high dual rank, since it is known that the uniquely colorable planar graphs are precisely the planar 3-trees. We then modify the semidefinite program to have an objective function with costs, and explore when we can create an objective function such that the optimal dual solution has sufficiently high rank. We show that it is always possible to construct such an objective function given the graph coloring. The construction of the objective function gives rise to heuristics for 4-coloring planar graphs. We enumerated all maximal planar graphs with an induced $$K_4$$ of up to 14 vertices; the heuristics successfully found a 4-coloring for 99.75\% of them. Our research was motivated by trying to use semidefinite programming to prove the four-color theorem, which states that every planar graph can be colored with four colors. There is an intriguing connection of the Karger-Motwani-Sudan semidefinite program with the Colin de Verdi\`ere graph invariant (and a corresponding conjecture of Colin de Verdi\`ere), in which matrices that have some similarities to the dual feasible matrices of the semidefinite program must have high rank in the case that graphs are of a certain type; for instance, planar graphs have rank that would imply that the primal solution of the semidefinite program encodes a 4-coloring. 

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4. 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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5. DP‐Coloring of Graphs From Random Covers

   https://doi.org/10.1002/rsa.70000  

   Bernshteyn, Anton; Dominik, Daniel; Kaul, Hemanshu; Mudrock, Jeffrey_A (February 2025, Random Structures & Algorithms)

   ABSTRACT DP‐coloring (also called correspondence coloring) of graphs is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. Intuitively, DP‐coloring generalizes list coloring by allowing the colors that are identified as the same to vary from edge to edge. Formally, DP‐coloring of a graph is equivalent to an independent transversal in an auxiliary structure called a DP‐cover of . In this paper, we introduce the notion of random DP‐covers and study the behavior of DP‐coloring from such random covers. We prove a series of results about the probability that a graph is or is not DP‐colorable from a random cover. These results support the following threshold behavior on random ‐fold DP‐covers as where is the maximum density of a graph: Graphs are non‐DP‐colorable with high‐probability when is sufficiently smaller than , and graphs are DP‐colorable with high‐probability when is sufficiently larger than . Our results depend on growing fast enough and imply a sharp threshold for dense enough graphs. For sparser graphs, we analyze DP‐colorability in terms of degeneracy. We also prove fractional DP‐coloring analogs to these results. 

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