An extremal graph for a given graph $H$ is a graph with maximum number of edges on fixed number of vertices without containing a copy of $H$. The $k$-th power of a path is a graph obtained from a path and joining all pair of vertices of the path with distance less than $k$. Applying a deep theorem of Simonovits, we characterize the extremal graphs of the $k$-th power of paths. This settles a conjecture posed by Xiao, Katona, Xiao and Zamora in a stronger form.
In this paper we deal with a Turán-type problem: given a positive integer n and a forbidden graph H, how many edges can there be in a graph on n vertices without a subgraph H? How does a graph look like if it has this extremal edge number? The forbidden graph in this article is a clique-path: a path of length k where each edge is extended to an r-clique, r >2. We determine both the extremal number and the extremal graphs for sufficiently large n.
Let $\overrightarrow{P_k}$ and $\overrightarrow{C_k}$ denote the directed path and the directed cycle of order $k$, respectively. In this paper, we determine the precise maximum size of $\overrightarrow{P_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size for large $n$. For all $n$, we also determine the precise maximum size of $\overrightarrow{C_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size. In addition, Huang and Lyu [\textit{Discrete Math. 343(5) 2020}] characterized the extremal digraphs avoiding an orient…
We study the eigenvectors of Laplacian matrices of trees. The Laplacian matrix is reduced to a tridiagonal matrix using the Schur complement. This preserves the eigenvectors and allows us to provide fomulas for the ratio of eigenvector entries. We also obtain bounds on the ratio of eigenvector entries along a path in terms of the eigenvalue and Perron values. The results are then applied to the Fiedler vector. Here we locate the extremal entries of the Fiedler vector and study classes of graphs such that the extremal entries can be found at the end points of the longest path.
We present an interactive framework that, given a membership test for a graph class $\mathcal{G}$ and a number $k$, finds and tests unavoidable sets for the class of graphs in $\mathcal{G}$ of path-width at most $k$. We put special emphasis on the case that $\mathcal{G}$ is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which yields highly efficient pruning techniques. Using this framework we determine all extremal girth values of cubic graphs of path-width $k$ for all $k \in \{3,\dots, 10\}…
Extremal problems on disjoint path covers of graphs
School of Mathematics and Statistics, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Key Laboratory of Analytical Mathematics and Applications (FJKLAMA), Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University（福建师范大学数学与统计学院，分析数学及应用教育部重点实验室，福建省分析数学及应用重点实验室（FJKLAMA），福建省应用数学中心（FJNU））
In this paper we define cycle-star graph CSk,n-k to be a graph on n vertices consisting of the cycle of length k and n-k leafs appended to the same vertex of the cycle. Also, we define cycle-path graph CPk,n-k to be a graph on n vertices consisting of the cycle of length k and of path on n-k vertices whose one end is linked to a vertex on a cycle. We establish that cycle-star graph CS3,n-3 is the only maximal graph with respect to additively weighted Harary index among all unicyclic graphs on n vertices, while cycle-path graph CP3,n-3 is the only minimal unicyclic graph (here n must be at leas…
A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph $G$, the \emph{monochromatic vertex-connection number}, denoted by $mvc(G)$, is defined to be the maximum number of colors used in an \emph{MVC-coloring} of $G$. These concepts of vertex-version are natural generalizations of the colorful monochromatic connect…
The \emph{blow-up} of a graph $H$ is the graph obtained from replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. Erdős et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blow-ups of paths. We determined the extremal number and found the extremal graphs for the blow-ups of cycles and a large class of trees, when $n$ is sufficiently large. This generalizes their results. The additional aim of our note is to draw attention to a powerful tool…
In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index τ(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.
For positive integers $n>d\geq k$, let $ϕ(n,d,k)$ denote the least integer $ϕ$ such that every $n$-vertex graph with at least $ϕ$ vertices of degree at least $d$ contains a path on $k+1$ vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function $ϕ(n,d,k)$, and conjectured that for any positive integers $n>d\geq k$, it holds that $ϕ(n,d,k)\leq \lfloor\frac{k-1}{2}\rfloor\lfloor\frac{n}{d+1}\rfloor+ε$, where $ε=1$ if $k$ is odd and $ε=2$ otherwise. In this paper we determine the values of the function $ϕ(n,d,k)$ exactly. This confirms the above conjecture…
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.
Many important problems in extremal combinatorics can be be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that hom$(H_1,G)^{a_1}\cdots$hom$(H_k,G)^{a_k}\geq$hom$(H_{k+1},G)^{a_{k+1}}\cdots$hom$(H_m,G)^{a_m}$ holds for some fixed graphs $H_1,\dots,H_m$ and all graphs $G$. One prominent example is Sidorenko's conjecture. For a fixed collection of graphs $\mathcal{U}=\{H_1,\dots,H_m\}$, the exponent vectors of valid pure binomial inequalities in graphs of $\mathcal{U}$ form a convex cone. We compute this cone for several families of graphs including com…
We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, η,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs $((x,y),(x,z))\in P\times Q$ with $\{x,y,z\}\in E$ is at least $d|\mathcal{K}_{P_2}(P,Q)|-η|V|^3,$ where $\mathcal{K}_{P_2}(P,Q)$ denotes the set of pairs in $P\times Q$ of the form $((x,y),(x,z))$. For a given $3$-uniform hypergraph $F$ we are interested in the infimum $d\geq 0$ such that for sufficiently small $η$ every sufficiently large $(d, η,P_2)$-dense hypergr…
The problem of determining the smallest number of edges, $h(n;\barκ\geq r)$, which guarantees that any graph with $n$ vertices and $h(n;\barκ\geq r)$ edges will contain a pair of vertices joined by $r$ internally disjoint paths was posed by Erdös and Gallai. Bollobás considered the problem of determining the largest number of edges $f(n;\barκ\leq \ell)$ for graphs with $n$ vertices and local connectivity at most $\ell$. One can see that $f(n;\barκ\leq \ell)= h(n;\barκ\geq \ell+1)-1$. These two problems had received a wide attention of many researchers in the last few decades. In the above prob…
Let $H_{\mathrm{WR}}$ be the path on $3$ vertices with a loop at each vertex. D. Galvin conjectured, and E. Cohen, W. Perkins and P. Tetali proved that for any $d$-regular simple graph $G$ on $n$ vertices we have $$\hom(G,H_{\mathrm{WR}})\leq \hom(K_{d+1},H_{\mathrm{WR}})^{n/(d+1)}.$$ In this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali. Our main tool is a simple bijection between the Widom-Rowlinson model and the hard-core model on another graph. We also give a large class of graphs $H$ for which we have $$\hom(G,H)\leq \hom(…
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $λ$ be an arbitrary integral weight. We denote by $\mathbb{B}(λ)$ the crystal of all Lakshmibai-Seshadri paths of shape $λ$. Let $V(λ)$ be the extremal weight module of extremal weight $λ$ generated by the (cyclic) extremal weight vector $v_λ$ of weight $λ$, and let $\mathcal{B}(λ)$ be the crystal basis of $V(λ)$ with $u_λ\in \mathcal{B}(λ)$ the element corresponding to $v_λ$. We prove that the connected component $\mathcal{B}_0(λ)$ of $\mathcal{B}(λ)$ containing $u_λ$ is isomorphic, as a crystal, to the connected compon…
One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Turán problem for (hyper-)graphs, can be encoded as special cases of thi…
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.
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers. We then determine the asymptotic order of the largest size of a graph with given general or monophonic position number, classifyin…
Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every $k$, we also determine the connected graphs that are extremal with respect to the number of $k$-vertex connected induced subgraphs. We show that, in contrast to the minimum which is…
Let $G$ be a graph with $n$ vertices and $λ_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal graphs. This result gives spectral conditions for the existence of specified paths and cycles in graphs.
A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.
A graph on $2k$ vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of length at most 3. In this paper we present an infinite family of path-pairable graphs with diameter $d(G)=O(\sqrt{n})$ where $n$ denotes the number of vertices of the graph. We prove that our example is extremal up to a constant factor.
Let G be a simple connected graph with n vertices, and let d_i be the degree of the vertex v_i in G. The extended adjacency matrix of G is defined so that the ij-entry is 1/2(d_i/d_j+d_j/d_i) if the vertices v_i and v_j are adjacent in G, and 0 otherwise. This matrix was originally introduced for developing novel topological indices used in the QSPR/QSAR studies. In this paper, we consider extremal problems of the largest eigenvalue of the extended adjacency matrix (also known as the extended spectral radius) of trees. We show that among all trees of order n>= 5, the path Pn(resp., the star Sn…
In this paper we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory -- we find explicit formulae for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to $(r,s)$ torus knots and show that their classical generating functions, in the extremal limit and framing $rs$, are generating functions of lattice paths under the line of the slope $r/s$. Generating functions of such paths satisfy extremal A-polynom…
The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $\lceil (2n-1)/3\rceil$ vertices and $\lfloor(n-2)/3\rfloor$ isolated vertices. For planar graphs, we show that the extremal $n$-vertex planar graph at…
Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we have $e \subset f(e)$. This generalizes the established definitions of "Berge path" and "Berge cycle" to general graphs. For a fixed graph $G$ we examine the maximum possible size (i.e.\ the sum of the cardinality of each edge) of a hypergraph with no Berge-$G$ as a subhypergraph. In the present paper we prove general bounds for this maximum when $G$ is an arbitr…
Given a graph $H$ and an integer $p$ ($p\geq 2$), the edge blow-up $H^{p+1}$ of $H$ is the graph obtained from replacing each edge in $H$ by a clique of order $(p+1)$, where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. Very recently some substantial progress of the extremal graphs for $H^{p+1}$ of larger $p$ has been made by Yuan. The range of Turán numbers for edge blow-up of all bipartite graphs when $p\geq 3$ and the exact Turán numbers for edge blow-up of all non-bipartite graphs when $p\geq χ(H) +1$…
We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding…
We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges. An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is also proved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars.
Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $ϕ: E(G)\to E(H)$ such that $e\subseteq ϕ(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the $p$-spectral radius $λ^{(p)}(H)$ of $H$ is defined as \[ λ^{(p)}(H):=\max_{{\bf x}\in\mathbb{R}^n,\,\|{\bf x}\|_p=1} r\sum_{\{i_1,i_2,\ldots,i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r}. \] In this paper, we study the $p$-spectral radius of Berge-$G$ hypergraphs. We determine the $3$-uniform hypergraphs with maximum $p$-spectral radius for $p\geq 1$ among Berge-$G$ hypergraph…
This article studies the \emph{$k$-forcing number} for oriented graphs, generalizing both the \emph{zero forcing number} for directed graphs and the $k$-forcing number for simple graphs. In particular, given a simple graph $G$, we introduce the maximum (minimum) oriented $k$-forcing number, denoted $\MOF_k(G)$ ($\mof_k(G)$), which is the largest (smallest) $k$-forcing number among all possible orientations of $G$. These new ideas are compared to known graph invariants and it is shown that, among other things, $\mof(G)$ equals the path covering number of $G$ while $\MOF_k(G)$ is greater than or…
Various questions related to distances between vertices of simple, finite graphs are of interest to extremal graph theorists. The Steiner distance of a set of $k$ vertices is a natural generalization of the regular distance. We extend several theorems on the middle parts and extremal values of trees from their regular distance variants to their Steiner distance variants. More specifically, we show that for a tree $T$, the Steiner $k$-distance, Steiner $k$-leaf-distance, and Steiner $k$-internal-distance are all concave along a path. We also calculate distances between the Steiner $k$-median, S…
The Erdős--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples $H_{n,k,t}$ and $H_{n,k,2}$. In this note, we generalize the result of Kopylov to bound the number of $s$-cliques in a graph with circumference less than $k$. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of…
