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Abstract We determine the maximum number of induced copies of a 5‐cycle in a graph on vertices for every . Every extremal construction is a balanced iterated blow‐up of the 5‐cycle with the possible exception of the smallest level where for , the Möbius ladder achieves the same number of induced 5‐cycles as the blow‐up of a 5‐cycle on eight vertices. This result completes the work of Balogh, Hu, Lidický, and Pfender, who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method, but we use a new and more sophisticated approach which allows us to extend its use to small graphs. 

Abstract For a plane near‐triangulation with the outer face bounded by a cycle , let denote the function that to each 4‐coloring of assigns the number of ways extends to a 4‐coloring of . The Block‐count reducibility argument (which has been developed in connection with attempted proofs of the Four Color Theorem) is equivalent to the statement that the function belongs to a certain cone in the space of all functions from 4‐colorings of to real numbers. We investigate the properties of this cone for , formulate a conjecture strengthening the Four Color Theorem, and present evidence supporting this conjecture. 

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. 

Abstract A graph $$H$$ is common if the number of monochromatic copies of $$H$$ in a 2-edge-colouring of the complete graph $$K_n$$ is asymptotically minimised by the random colouring. Burr and Rosta, extending a famous conjecture of Erdős, conjectured that every graph is common. The conjectures of Erdős and of Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples of common graphs had not seen much progress since then, although very recently a few more graphs were verified to be common by the flag algebra method or the recent progress on Sidorenko’s conjecture. Our contribution here is to provide several new classes of tripartite common graphs. The first example is the class of so-called triangle trees, which generalises two theorems by Sidorenko and answers a question of Jagger, Šťovíček, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree $$T$$ , there exists a triangle tree such that the graph obtained by adding $$T$$ as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most $$5$$ vertices yields a common graph. 

The planar Turán number $$\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$$ is the largest number of edges in an $$n$$-vertex planar graph with no $$\ell$$-cycle. For each $$\ell\in \{3,4,5,6\}$$, upper bounds on $$\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$$ are known that hold with equality infinitely often. Ghosh, Győri, Martin, Paulos, and Xiao [arXiv:2004.14094] conjectured an upper bound on $$\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$$ for every $$\ell\ge 7$$ and $$n$$ sufficiently large. We disprove this conjecture for every $$\ell\ge 11$$. We also propose two revised versions of the conjecture. 

If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries $$\pm1$$, then the matrix formed by taking the columns as the eigenvectors is a Hadamard matrix and the graph is said to be Hadamard diagonalizable. In this article, we prove that if $n=8k+4$ the only possible Hadamard diagonalizable graphs are $$K_n$$, $$K_{n/2,n/2}$$, $$2K_{n/2}$$, and $$nK_1$$, and we develop a computational method for determining all graphs diagonalized by a given Hadamard matrix of any order. Using these two tools, we determine and present all Hadamard diagonalizable graphs up to order 36. Note that it is not even known how many Hadamard matrices there are of order 36.