For a graph $H$, the 1-subdivision of $H$, denoted by $H'$, is the graph obtained by replacing the edges of $H$ by internally disjoint paths of length 2. Recently, Conlon, Janzer and Lee (arXiv: 1903.10631) asked the following question: For any integer $s\ge2$, estimate the smallest $t$ such that $\textup{ex}(n,K_{s,t}')=Ω(n^{\frac{3}{2}-\frac{1}{2s}})$. In this paper, we consider the case $s=3$. More precisely, we provide an explicit construction giving \begin{align*} \text{ex}(n,K_{3,30}')=Ω(n^{\frac{4}{3}}), \end{align*} which reduces the estimation for the smallest value of $t$ from a magnitude of $10^{56}$ to the number $30$. The construction is algebraic, which is based on some equations over finite fields.
We show that for any integer $t\geq 2$, every properly edge-coloured graph on $n$ vertices with more than $n^{1+o(1)}$ edges contains a rainbow subdivision of $K_t$. Note that this bound on the number of edges is sharp up to the $o(1)$ error term. This is a rainbow analogue of some classical results on clique subdivisions and extends some results on rainbow Turán numbers. Our method relies on the framework introduced by Sudakov and Tomon[2020] which we adapt to find robust expanders in the coloured setting.
Given a graph $H$ and a positive integer $n$, the {\it Turán number} $\ex(n,H)$ is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. A real number $r\in(1,2)$ is called a {\it Turán exponent} if there exists a bipartite graph $H$ such that $\ex(n,H)=Θ(n^r)$. A long-standing conjecture of Erdős and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q> p$. In this paper, we build on recent developments on the conjecture to establish a large family of new Turán exponents. In particular, it follows from o…
We show that for every graph $G$ and every graph $H$ obtained by subdividing each edge of $G$ at least $O(\log |V(G)|)$, $H$ is nonrepetitively 3-colorable. In fact, we show that $O(\log π'(G))$ subdivisions per edge are enough, where $π'(G)$ is the nonrepetitive chromatic index of $G$. This answers a question of Wood and improves a similar result of Pezarski and Zmarz that stated the existence of at least one 3-colorable division with a linear number of subdivision vertices per edge.
The $h$-polynomial of the barycentric subdivision of any $n$-dimensional cubical complex with nonnegative cubical $h$-vector is shown to have only real roots and to be interlaced by the Eulerian polynomial of type $B_n$. This result applies to barycentric subdivisions of shellable cubical complexes and, in particular, to barycentric subdivisions of cubical convex polytopes and answers affirmatively a question of Brenti, Mohammadi and Welker.
The cubical barycentric subdivision sd_c(K) of a cubical complex K is introduced as an analogue of the barycentric subdivision of a simplicial complex. Explicit formulas for the short and long cubical h-vector of sd_c(K) are given, in terms of those of K. It is deduced that symmetry and nonnegativity of these h-vectors, as well as real rootedness of the short cubical h-polynomial, are preserved under cubical barycentric subdivision. The asymptotic behavior of the short and long cubical h-vectors of successive cubical barycentric subdivisions of K is also determined.
For a multigraph $F$, the $k$-subdivision of $F$ is the graph obtained by replacing the edges of $F$ with pairwise internally vertex-disjoint paths of length $k+1$. Conlon and Lee conjectured that if $k$ is even, then the $(k-1)$-subdivision of any multigraph has extremal number $O(n^{1+\frac{1}{k}})$, and moreover, that for any simple graph $F$ there exists $\varepsilon>0$ such that the $(k-1)$-subdivision of $F$ has extremal number $O(n^{1+\frac{1}{k}-\varepsilon})$. In this paper, we prove both conjectures.
Bimonotone subdivisions in two dimensions are subdivisions all of whose sides are either vertical or have nonnegative slope. They correspond to statistical estimates of probability distributions of strongly positively dependent random variables. The number of bimonotone subdivisions compared to the total number of subdivisions of a point configuration provides insight into how often the random variables are positively dependent. We give recursions as well as formulas for the numbers of bimonotone and total subdivisions of $2\times n$ grid configurations in the plane. Furthermore, we connect th…
In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G \circ P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of characterization of trees with the domination subdivision number equal to 3.
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).
The \textit{packing chromatic number} of a graph $G$, denoted by $% χ_ρ(G)$, is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we present some general properties of packing chromatic numbers of \textit{finite super subdivisions} of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and \textit{neighborhood corona graphs} of a cycle and a path respectively of a complete graph and a path.
Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the total dominator chromatic number of $k$-subdivision of $G$.
The \emph{domination subdivision number} sd$(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of $G$. It has been shown \cite{vel} that sd$(T)\leq 3$ for any tree $T$. We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the \emph{domination multisubdivision number} of a nonempty graph $G$ as a minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to incr…
Let $H_t$ be the subdivision of $K_t$. Very recently, Conlon and Lee have proved that for any integer $t\geq 3$, there exists a constant $C$ such that $\text{ex}(n,H_t)\leq Cn^{3/2-1/6^t}$. In this paper, we prove that there exists a constant $C'$ such that $\text{ex}(n,H_t)\leq C'n^{3/2-\frac{1}{4t-6}}$.
The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by subdividing each edge exactly once). * For every c > 0, every graph of sufficiently large average degree contains as a subgraph the 1-subdivision of…
An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. A vertex colouring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-free graph colouring was independently introduced by Kamčev, Łuczak and Sudakov and ourselves. In this paper we introduce the study of anagram-free colourings of graph subdivisions. We show that every graph has an anagram-free $8$-colourable subdivision. The number of division vertices per edge is exponential in the number of edges. For trees, we construct anagram-free $10$-colourable subdivisions wi…
The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every $n$-vertex graph with at least $\varepsilon n^2$ edges contains a $1$-subdivision of the complete graph on $c_{\varepsilon}\sqrt{n}$ vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least $(2+o(1))k^2$ vertices contains th…
The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers tha…
We prove that for every digraph $F$ and every assignment of pairs of integers $(r_e,q_e)_{e \in A(F)}$ to its arcs there exists an integer $N$ such that every digraph $D$ with dichromatic number at least $N$ contains a subdivision of $F$ in which $e$ is subdivided into a directed path of length congruent to $r_e$ modulo $q_e$, for every $e \in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result.
How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on $n$ vertices with a forbidden clique subdivision or immersion. We prove for $t$ sufficiently large that every graph on $n \geq t$ vertices with no $K_t$-immersion has at most $2^{t+\log^2 t}n$ cliques, which is sharp apart from the $2^{O(\log^2 t)}$ factor. We also prove that the maximum number of cliques in an $n$-vertex graph with no $K_t$-subdivision is at most $2^{1.817t}n$.…
We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to zero as $t$ tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in $n$-vertex graphs with no $K_t$-minor is at most $2^{ct}n$ for some constant $c$.
For a graph $F$, the $k$-subdivision of $F$, denoted $F^k$, is the graph obtained by replacing the edges of $F$ with internally vertex-disjoint paths of length $k$. In this paper, we prove that $\mathrm{ex}(n,K_{s,t}^k)=O(n^{1+\frac{s-1}{sk}})$, which is tight for $t$ sufficiently large. This settles a conjecture of Conlon--Janzer--Lee, and improves on a substantial body of work by Conlon--Janzer--Lee and Jiang--Qiu.
In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we…
We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vecχ}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $\text{mader}_{\vecχ}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is tha…
Let $K_4^+$ be the 5-vertex graph obtained from $K_4$, the complete graph on four vertices, by subdividing one edge precisely once (i.e. by replacing one edge by a path on three vertices). We prove that if the chromatic number of some graph $G$ is much larger than its clique number, then $G$ contains a subdivision of $K_4^+$ as an induced subgraph.
Given a graph $H$, the Turán number $ex(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a recent conjecture of Conlon, Janzer, and Lee on the Turán numbers of bipartite graphs, which in turn yields further progress on a conjecture of Erdős and Simonovits. Let $s,t,k\geq 2$ be integers. Let $K_{s,t}^k$ denote the graph obtained from the complete bipartite graph $K_{s,t}$ by replacing each edge $uv$ in it with a path of length $k$ between $u$ and $v$ such that the $st$ replacing paths are internally disjoint. It follows from a general theorem of Buk…
In this paper we analyze a variant of the pursuit-evasion game on a graph $G$ where the intruder occupies a vertex, is allowed to move to adjacent vertices or remain in place, and is 'invisible' to the searcher, meaning that the searcher operates with no knowledge of the position of the intruder. On each stage, the searcher is allowed to inspect an arbitrary set of $k$ vertices. The minimum $k$ for which the searcher can guarantee the capture of the intruder is called the inspection number of $G$. We also introduce and study the topological inspection number, a quantity that captures the limit…
For a graph $G$, let $χ(G)$ denote its chromatic number and $σ(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $χ(G)/σ(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Hajós from 1961 states that $σ(G) \geq χ(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erdős and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in t…
Let $e$ be an edge of a connected simple graph $G$. The graph obtained by removing (subdividing) an edge $e$ from $G$ is denoted by $G-e$ ($G_e$). As usual, $γ(G)$ denotes the domination number of $G$. We call $G$ an SR-graph if $γ(G-e) = γ(G_e)$ for any edge $e$ of $G$, and $G$ is an ASR-graph if $γ(G - e) \neq (G_e)$ for any edge $e$ of $G$. In this work we give several examples of SR and ASR-graphs. Also, we characterize SR-trees and show that ASR-graphs are $γ$-insensitive.
We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. The bound depends only on the number of subdivisions, and we construct examples which prove that it is sharp.
We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative answer to a question of W. Kuehnel whether every 4-connected graph G contains a subdivision H of K4 as a subgraph such that G-V(H) is connected. The motor for our induction is a result of Fontet and Martinov stating that every 4-connected graph can be reduced to a smaller one by contracting a single edge, unless the graph is the square of a cycle or the line …
Let $H$ be a graph with maximum degree $d$, and let $d'\ge 0$. We show that for some $c>0$ depending on $H,d'$, and all integers $n\ge 0$, there are at most $c^n$ unlabelled simple $d$-connected $n$-vertex graphs with maximum degree at most $d'$ that do not contain $H$ as a subdivision. On the other hand, the number of unlabelled simple $(d-1)$-connected $n$-vertex graphs with minimum degree $d$ and maximum degree at most $d+1$ that do not contain $K_{d+1}$ as a subdivision is superexponential in $n$.
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the tropical intersections problem derived from the Kuramoto equations. Regular subdivisions of adjacency polytopes are instrumental in solving these problems. This paper explores connections between the regular subdivisions of an adjacency polytope and the contraction of the underlying graph along an edge. We construct a special …
In 1984, Thomassen conjectured that for every constant $k \in \mathbb{N}$, there exists $d$ such that every graph with average degree at least $d$ contains a balanced subdivision of a complete graph on $k$ vertices, i.e. a subdivision in which each edge is subdivided the same number of times. Recently, Liu and Montgomery confirmed Thomassen's conjecture. We show that for every constant $0<c<1/2$, every graph with average degree at least $d$ contains a balanced subdivision of a complete graph of size at least $Ω(d^{c})$. Note that this bound is almost optimal. Moreover, we show that every spars…
We call a set $\mathcal S$ of graphs an "even subdivison-factor" of a cubic graph $G$ if $G$ contains a spanning subgraph $H$ such that every component of $H$ has an even number of vertices and is a subdivision of an element of $\mathcal S$. We show that any set of 2-connected graphs which is an even subdivison-factor of every 3-connected cubic graph, satisfies certain properties. As a consequence, we disprove a conjecture which was stated in an attempt to solve the circuit double cover conjecture.
Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color isomorphic copies of $H$. In this paper, we prove that $f_{2}(n,H_{t})=Ω(n^{1+\frac{1}{2t-3}})$ where $H_{t}$ is the $1$-subdivision of the complete graph $K_{t}$. This answers a question of Conlon and Tyomkyn (arXiv: 2002.00921).
The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest number of edges a graph $G$ can have, such that for every edge-coloring of $G$ with $r$ colors there exists a monochromatic copy of $H$ in $G$. For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q{-}1$ vertices each. In a recent paper of Kohayakawa, Retter and R{ö}dl, it is shown that for all constant integers $q,r\geq 2$ and every graph $H$ on $n$ vertices and of bounded maximum degree, the $r$-size-Ramsey number of $H^q$ is at most $(\log n)^{20(q-1)}n^{1+1/q}$, for $n$ large …
A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of M is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen sub…
A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$ vertices. We prove that there are $Ω(n^3)$ such $K_4$-subdivisions and show that the order of this bound is tight for infinitely many graphs. We further investigate a better bound in dependence on $m$ and prove that the computational complexity of the problem of counting the exact number of $K_4$-subdivisions is $\#P$-hard.
Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing $K'_{s,t}$ for the subdivision of the bipartite graph $K_{s,t}$, we show that $\mathrm{ex}(n, K'_{s,t}) = O(n^{3/2 - \frac{1}{2s}})$. This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for $t$ sufficiently large in terms of $s$. Second, for any integers $s, k \geq 1$, we show that $\mathrm{ex}(n, L) = Θ(n^{1 + \frac{s}{sk+1}})$ fo…
Given a graph $H$, a balanced subdivision of $H$ is a graph obtained from $H$ by subdividing every edge the same number of times. In 1984, Thomassen conjectured that for each integer $k\ge 1$, high average degree is sufficient to guarantee a balanced subdivision of $K_k$. Recently, Liu and Montgomery resolved this conjecture. We give an optimal estimate up to an absolute constant factor by showing that there exists $c>0$ such that for sufficiently large $d$, every graph with average degree at least $d$ contains a balanced subdivision of a clique with at least $cd^{1/2}$ vertices. It also confi…
For two simple graphs $G_1$ and $G_2$, we denote the subdivision-vertex join and subdivision-edge join of $G_1$ and $G_2$ by $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$, respectively. This paper determines the normalized Laplacian spectra of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ in terms of these of $G_1$ and $G_2$ whenever $G_1$ and $G_2$ are regular. As applications, we construct some non-regular normalized Laplacian cospectral graphs. Besides we also compute the number of spanning trees and the degree-Kirchhoff index of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ for regular graphs $G_1$ and $G_2…
A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph $K_4$. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-r…
Inspired by recent results of Ein, Lazarsfeld, Erman and Zhou on the non-vanishing of Betti numbers of high Veronese subrings, we describe the behaviour of the Betti numbers of Stanley-Reisner rings associated with iterated barycentric or edgewise subdivisions of a given simplicial complex. Our results show that for a simplicial complex $Δ$ of dimension $d-1$ and for $1\leq j\leq d-1$ the number of $0$'s the j-th linear strand of the minimal free resolution of the r-th barycentric or edgewise subdivision is bounded above only in terms of $d$ and $j$ (and independently of $r$).
