In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Turán problems where the host graph is planar. In particular we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geq 1$. We obtain the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.
The Cartesian product of P_2 and P_n is called an n-ladder graph for a positive integer n. We call two paths P_m and P_n together with some edges each of which joins a vertex on P_m and a vertex on P_n a generalized (m,n)-ladder graph. In this paper, we completely characterize the planar generalized ladder graphs and the outerplanar generalized ladder graphs. A functigraph C(P_n ,f) is a generalized (n,n)-ladder graph. Consequently, our result solves the problem posed by A. Chen et al. (2011) to characterize planar functigraphs C(P_n ,f).
The {\em planar Turán number} of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $H$-free planar graph. The planar Turán number of $k\geq 3$ vertex-disjoint union of cycles is a trivial value $3n-6$. Lan, Shi and Song determine the exact value of $ex_{\mathcal{P}}(n,2C_3)$. We continue to study planar Turán number of vertex-disjoint union of cycles and obtain the exact value of $ex_{\mathcal{P}}(n,H)$, where $H$ is vertex-disjoint union of $C_3$ and $C_4$. The extremal graphs are also characterized. We also improve the lower bound of $ex_{\mathcal{P}}(n,2C_k)$ whe…
HUN-REN, Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Department of Computer Science and Information Theory, Faculty of Electrical Engineering and Informatics, Budapest University of Technology and Economics（布达佩斯技术与经济大学电气与信息工程学院计算机科学与信息论系）
The class of fighting fish is a recently introduced model of branching surfaces generalizing parallelogram polyominoes. We can alternatively see them as gluings of cells, walks on the square lattice confined to the quadrant or shuffle of Dyck words. With these different points of view, we introduce a natural extension of fighting fish that we call \emph{generalized fighting fish}. We show that generalized fighting fish are exactly the Mullin codes of rooted planar maps endowed with their unique rightmost depth-first search spanning tree, also known as Lehman-Lenormand code. In particular, this…
The planar Turán number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph, which is obtained from $k$ triangles by sharing a common vertex. In this paper, we obtain sharp bounds of $ex_{\mathcal{P}}(n,H_k)$ and $ex_{\mathcal{P}}(n,K_1+P_{k+1})$ for $k\ge2$, which improves the results of Lan and Shi in Electron. J. Combin. 26 (2) (2019), \#P2.11.
Young's lattice is a prototypical example of differential posets. Differential posets have the Robinson correspondence, the correspondence between permutations and pairs of standard tableaux with the same shape, as in the case of Young's lattice. Fomin introduced generalized Schur operators to generalize the method of Robinson correspondence in differential posets to the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard tableaux with the same shape. In this paper, we introduce operators on the vector space whose basis is the set of …
The planar Turan 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 $\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ are known that hold with equality infinitely often. Ghosh, Györi, Martin, Paulo, 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.
An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$, known as the rainbow Turán problem, was initiated by Keevash, Mubayi, Sudakov and Verstraëte. We investigate a variation of this problem with the additional restriction that the graph is planar, and we denote the corresponding extremal number by $\ex_{\p}^*(n,F)$. In particular, we determine $\ex_{\p}^*(n,P_5)$, where $P_5$ denotes the $5$-vertex path.
The planar Turán number of a graph $H$, denoted $ex_{_\mathcal{P}}(n,H)$, is the maximum number of edges in a planar graph on $n$ vertices without containing $H$ as a subgraph. This notion was introduced by Dowden in 2016 and has attracted quite some attention since then; those work mainly focus on finding $ex_{_\mathcal{P}}(n,H)$ when $H$ is a cycle or Theta graph or $H$ has maximum degree at least four. In this paper, we study $ex_{_\mathcal{P}}(n,H)$ when $H$ is a cubic graph or disjoint union of cycles or $H=K_{s, t}$.
Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well studied function, the planar Turán number of $H$, denoted by ${\rm ex}_{\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${\rm ex}_{\mathcal{P}}(n,C_4)$ and ${\rm ex}_{\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In thi…
Given a graph $F$, the planar Turán number of $F$, denoted $\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining sharp upper bound for $\text{ex}_{\mathcal{P}}(n,C_4)$ and $\text{ex}_{\mathcal{P}}(n,C_5)$, where $C_4$ and $C_5$ are cycles of length four and five respectively. In this paper we determined an upper bound for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$, $\text{ex}_{\mathcal{P}}(n,S_{2,3})$, $\text{ex}_{\mathcal{P}}(n,S_{2,4})$, $\text{ex}_{\mathcal{P}}(n,S_{2,5})$, $…
The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give addition…
The planar Turán number of a graph $H$, denoted by $ex_{_\mathcal{P}}(n,H)$, is the largest number of edges in a planar graph on $n $ vertices without containing $H$ as a subgraph. In this paper, we continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230]. We first obtain an improved lower bound for $ex_{_\mathcal{P}}(n,C_k)$ for all $k\ge 13$ and $n\ge 5(k-6+\lfloor{(k-1)}/2\rfloor)(k-1)/2$; the construction for each $k$ and $n$ provides a simpler counterexample to a conjecture of Ghosh, Győri, Martin, Paulos and Xiao [arxiv:2004.14094v1…
We study Koebe orderings of planar graphs: vertex orderings obtained by modelling the graph as the intersection graph of pairwise internally-disjoint discs in the plane, and ordering the vertices by non-increasing radii of the associated discs. We prove that for every $d\in \mathbb{N}$, any such ordering has $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring number bounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibility of planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches a known lower bound due to Dvořák and Siebertz.
Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer $n$ and a planar graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains a subgraph isomorphic to $H$. The planar anti-Ramsey number of $H$, denoted $ar_{_\mathcal{P}}(n, H)$, is the maximum number of colors in an edge-coloring of a plane triangulation $T\in \mathcal{T}_n(H)$ such that $T$ contains no rainbow copy of $H$. Analogous to anti-Ramsey numbe…
An improved upper bound for the planar Turán number of $C_8$
School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；School of Mathematics, Shandong University（山东大学数学学院）；Data Science Institute, Shandong University（山东大学数据科学研究院）
A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron correspon…
Thomassen in 1994 published a famous proof of the fact that the choosability of a planar graph is at most 5. Zhu in 2019 generalized this result by showing that the same bound holds for Alon-Tarsi numbers of planar graphs. We present an alternative proof of that fact, derived from the results on decompositions of planar graphs into trees known as Schnyder woods. It turns out that Thomassen's technique and our proof based on Schnyder woods have a lot in common. We discuss and explain the prominent role that counterclockwise 3-orientations play in proofs based on both these approaches.
In recent work of Cachazo, Guevara, Mizera and the author, a generalization of the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ was introduced as an integral over the moduli space of $n$ points in $\mathbb{CP}^{k-1}$, with value a sum of certain rational functions on the kinematic space $\mathcal{K}_{k,n}$. It was shown there for $m^{(3)}(\mathbb{I}_6,\mathbb{I}_6)$ and later by Cachazo and Rojas that collections of poles appearing in $m^{(3)}(\mathbb{I}_7,\mathbb{I}_7)$ are compatible exactly when they are dual to collections of rays which generate the maximal faces of …
It is known that DP-coloring is a generalization of a list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-coloring which is a generalization if a relaxed list coloring. We also shows that every planar graph $G$ without $4$-cycles or $6$-cycles is DP-$(k,d)^*$-colorable. It follows immediately that $G$ is $(k,d)^*$-choosable.
We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle the generalized outerplanar Turán number for all cycles. We also determine the exponential growth of the generalized outerplanar Turán number of paths Pk as a function of k which implies the order of magnitude of the generalized outerplanar Turán number of arbitrary trees. The bounds are strongly related to the sequence of Catalan numbers.
Given two graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in an $n$-vertex $F$-free graph. For every $F$ and sufficiently large $n$, we present an extremal graph for a generalized Turán problem, i.e., an $F$-free $n$ vertex graph $G$ that for some $H$ contains exactly $\mathrm{ex}(n,H,F)$ copies of $H$.
In this paper we estimate the planar Turán number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when $H=C_5$, and study the cases when $G$ is bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly new applications or variants of the "contribution method" introduced by Ghosh, Győri, Martin, Paulos and Xiao in arXiv:2004.14094.
For graphs $H$ and $F$, the generalized Turán number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We consider this problem when both $H$ and $F$ have at most four vertices. We give sharp results in almost all cases, and connect the remaining cases to well-known unsolved problems. Our main new contribution is applying the progressive induction method of Simonovits for generalized Turán problems.
Given graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains almost $\mathrm{ex}(n,H,F)$ copies of $H$, than $G$ is in some sense similar to some extremal graph. We obtain new stability results for generalized Turán problems and derive several new exact results.
We study the generalized Turán function $ex(n,H,F)$, when $H$ or $F$ is $K_{2,t}$. We determine the order of magnitude of $ex(n,H,K_{2,t})$ when $H$ is a tree, and determine its asymptotics for a large class of trees. We also determine the asymptotics of $ex(n,K_{2,t},H)$ in most cases.
For graph $G$, $F$ and integer $n$, the generalized Tuán number $ex(n,G,F)$ denotes the maximum number of copies of $G$ that an $F$-free $n$-vertex graph can have. We study this parameter when both $G$ and $F$ are complete bipartite graphs.
We prove a linear recurrence relation for a large family of generalized Schreier sets, which generalizes the Fibonacci recurrence proved by Bird and higher order Fibonacci recurrence proved by the second author et al. Furthermore, we show a relationship between Schreier-type sets and Turán graphs.
We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum-product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of $\max\{|A+A|,|AA|\}$ is neither close to $|A|$ nor to $|A|^2$. To obtain our main sum-product bound, we extend Elekes' sum-product technique that relies on point-line…
We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms.
It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an $n$-vertex planar graph, augments this graph in $O(n^2)$ steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized Antifactor-problem for the fixed embedding case, and SPQR-trees for the variab…
We initiate a systematic study of non-planar on-shell diagrams in N=4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the dlog form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in th…
