Let $r>2$ and $σ\in(0,r-1)$ be integers. We require $t<2s$, where $t=2^{σ+1}-1$ and $s=2^{r-σ-1}$. Generalizing a known $\{K_4,T_{6,3}\}$-ultrahomogenous graph $G_3^1$, we find that a finite, connected, undirected, arc-transitive graph $G_r^σ$ exists each of whose edges is shared by just two maximal subgraphs, namely a clique $X_0=K_{2s}$ and a $t$-partite regular-Turán graph $X_1=T_{st,t}$ on $s$ vertices per part. Each copy $Y$ of $X_i$ ($i=0,1$) in $G_r^σ$ shares each edge with just one copy of $X_{1-i}$ and all such copies of $X_{1-i}$ are pairwise distinct. Moreover, $G_r^σ$ is an edge-disjoint union of copies of $X_i$, for $i=0,1$. We prove that $G_r^σ$ is $\{K_{2s},T_{st,t}\}$-homogeneous if $t<2s$, and just $\{T_{st,t}\}$-homogeneous otherwise, meaning that there is an automorphism of $G_r^σ$ between any two such copies of $X_i$ relating two preselected arcs.
We prove an infinitary disjoint union theorem for level products of trees. To implement the proof we develop a Hales-Jewett type result for words indexed by a level product of trees.
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…
A graceful difference labeling (gdl for short) of a directed graph G with vertex set V is a bijection f between V and {1,...,|V|} such that, when each arc uv is assigned the difference label f(v)-f(u), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.
Equitable $k$-choosability is a list analogue of equitable $k$-coloring that was introduced by Kostochka, Pelsmajer, and West in 2003. It is known that if vertex disjoint graphs $G_1$ and $G_2$ are equitably $k$-choosable, the disjoint union of $G_1$ and $G_2$ may not be equitably $k$-choosable. Given any $m \in \mathbb{N}$ the values of $k$ for which $K_{1,m}$ is equitably $k$-choosable are known. Also, a complete characterization of equitably $2$-choosable graphs is not known. With these facts in mind, we study the equitable choosability of $\sum_{i=1}^n K_{1,m_i}$, the disjoint union of $n$…
We prove that if $G$ is a $2r$-regular edge graceful $(p,q)$ graph with $(r,kp)=1$ then $kG$ is edge graceful for odd $k$. We also prove that for certain specific classes of $2r$-regular edge graceful graphs it is possible to drop the requirement that $(r,kp)=1$
If X is an n-element set, we call a family G of subsets of X a k-generator for X if every subset of X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo conjectured that for n > 2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjecture for all sufficiently large n when k = 2, and for n a sufficiently large multiple of k when k > 2.
A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = αA + βI + γJ + δD$ with $α\neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the diagonal matrix with the vertex degrees. In the case that $G$ is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix o…
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
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 $H=K_{s, t}$.
We call a family G of subsets of [n] a k-generator of (\mathbb{P}[n]) if every (x \subset [n]) can be expressed as a union of at most k disjoint sets in (\mathcal{G}). Frein, Leveque and Sebo conjectured that for any (n \geq k), such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize a theorem of Alon and Frankl \cite{alon} in order to show that for fixed k, any k-generator of (\mathbb{P}[n]) must have size at least (k2^{n/k}(1-o(1))), thereb…
Given $k$ pairs of vertices $(s_i,t_i)\;(1\le i\le k)$ of a digraph $G$, how can we test whether there exist vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but in an earlier paper we proved that for all fixed $k$, there is a polynomial-time algorithm to solve the problem if $G$ is a tournament (or more generally, a semicomplete digraph). Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $V(G)$ is partitioned into a bounded number of sets each inducing a semicomple…
The Turán number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices of a cycle $C_{n-1}$. Let $mW_{2k+1}$ denote the $m$ vertex-disjoint copies of $W_{2k+1}$. For sufficiently large $n$, we determine the Turán number and all extremal graphs for $mW_{2k+1}$. We also provide the Turán number and all extremal graphs for $W^{h}:=\bigcup\limits^m_{i=1}W_{k_i}$ when $n$ is sufficiently large, where the number of even wheels is $h$ a…
Recently, Chang and Hyun obtained some classes of binary optimal codes via simplicial complexes. In this letter, we utilize posets of the disjoint union of two chains to construct binary optimal linear codes.
The following conjecture was proposed in 2010 by S. Lando. Let M and N be two unions of the same number of disjoint circles in a sphere. Then there exist two spheres in 3-space whose intersection is transversal and is a union of disjoint circles that is situated as M in one sphere and as N in the other . Define union A of disjoint circles to be situated in one sphere as union B of disjoint circles in the other sphere if there is a homeomorphism between these two spheres which maps A to B. In this paper we prove that there exists pair of sets of 7 circles in sphere, that is a counterexample to …
In this paper we formulate cycle-supermagic labelings for the disjoint union of isomorphic copies of different families of graphs. We also prove that disjoint union of non isomorphic copies of fans and ladders are cycle-supermagic.
Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any coloring of the edges of $K_N$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $χ(G)$ be the chromatic number of $G$. Let $s(G)$ denote the chromatic surplus of $G$, the cardinality of a minimum color class taken over all proper colorings of $G$ with $χ(G)$ colors. Burr showed that for a connected graph $G$ and a graph $H$ with $v(G)\geq s(H)$, $R(G,H) \geq (v(G)-1)(χ(H)-1)+s(H)$. A connected graph $G$ is called $H$-good if $R(G,H)=(v(G)-1)(χ(H)-1)+s(…
We study extremal type problem arising from the question: What is the maximum number of edge-disjoint non-crossing perfect matchings on a set S of 2n points in the plane such that their union is a triangle-free geometric graph? We approach this problem by considering four different situations of S. In particular, in the general position, we obtain (i) a sufficient condition for the existence of n edge-disjoint non-crossing perfect matchings in the general position whose union is a maximal triangle-free geometric graph, and (ii) a lower bound on the number of edge-disjoint non-crossing perfect …
We prove that any $n$-vertex graph whose complement is triangle-free contains $n^2/12-o(n^2)$ edge-disjoint triangles. This is tight for the disjoint union of two cliques of order $n/2$. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erdős.
Dirac and Lovász independently characterized the $3$-connected graphs with no pair of vertex-disjoint cycles. Equivalently, they characterized all $3$-connected graphs with no prism-minors. In this paper, we completely characterize the $3$-connected graphs with an edge that is contained in the union of no pair of vertex-disjoint cycles. As applications, we answer the analogous questions for edge-disjoint cycles and for $4$-connected graphs and we completely characterize the $3$-connected graphs with no prism-minor using a specified edge.
Two new techniques are introduced into the theory of the domination game. The cutting lemma bounds the game domination number of a partially dominated graph with the game domination number of suitably modified partially dominated graph. The union lemma bounds the S-game domination number of a disjoint union of paths using appropriate weighting functions. Using these tools a conjecture asserting that the so-called three legged spiders are game domination critical graphs is proved. An extended cutting lemma is also derived and all game domination critical trees on 18, 19, and 20 vertices are lis…
Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)…
We consider the problem of determining the maximum order of an induced vertex-disjoint union of cliques in a graph. More specifically, given some family of graphs $\mathcal{G}$ of equal order, we are interested in the parameter $a(\mathcal{G}) = \min_{G \in \mathcal{G}} \max \{ |U| : U \subseteq V, G[U] \text{ is a vertex-disjoint union of cliques} \}$. We determine the value of this parameter precisely when $\mathcal{G}$ is the family of comparability graphs of $n$-element posets with acyclic cover graph. In particular, we show that $a(\mathcal{G}) = (n+o(n))/\log_2 (n)$ in this class.
We show that if $f\colon S_n \to \{0,1\}$ is $ε$-close to linear in $L_2$ and $\mathbb{E}[f] \leq 1/2$ then $f$ is $O(ε)$-close to a union of "mostly disjoint" cosets, and moreover this is sharp: any such union is close to linear. This constitutes a sharp Friedgut-Kalai-Naor theorem for the symmetric group. Using similar techniques, we show that if $f\colon S_n \to \mathbb{R}$ is linear, $\Pr[f \notin \{0,1\}] \leq ε$, and $\Pr[f = 1] \leq 1/2$, then $f$ is $O(ε)$-close to a union of mostly disjoint cosets, and this is also sharp; and that if $f\colon S_n \to \mathbb{R}$ is linear and $ε$-clos…
Let $c\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\lfloor 1/c \rfloor$ paths whose union covers all but at most $o(n)$ vertices of $G$. The constant $\lfloor 1/c \rfloor$ is best possible when $1/c\notin \mathbb{N}$ and off by $1$ otherwise. Moreover, if in addition $G$ is bipartite, then the number of paths can be reduced to $\lfloor 1/(2c) \rfloor$, which is best possible.
The $kK_{r+1}$ is the union of $k$ disjoint copies of $(r+1)$-clique. Moon [Canad. J. Math. 20 (1968) 95--102] and Simonovits [Theory of Graphs (Proc. colloq., Tihany, 1996)] independently showed that if $n$ is sufficiently large, then $K_{k-1}\vee T_{n-k+1,r}$ is the unique extremal graph for $kK_{r+1}$. In this paper, we consider the graph which has the maximum spectral radius among all graphs without $k$ disjoint cliques. We prove that if $G$ attains the maximum spectral radius over all $n$-vertex $kK_{r+1}$-free graphs for sufficiently large $n$, then $G$ is isomorphic to $K_{k-1}\vee T_{n…
Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is best possible and the degree condition is asymptotically best possible. This result also has an interpretation as the \emph{deficiency problems}, recently introduced by Nenadov, Sudakov and Wagner: every such $H$ can be made Hamiltonian by adding at most two vertices and all triples intersecting them.
An edge-colored graph is called rainbow if all the colors on its edges are distinct. Given a positive integer n and a graph G, the anti-Ramsey number ar(n,G) is the maximum number of colors in an edge-coloring of K_{n} with no rainbow copy of G. Denote by kC_{3} the union of k vertex-disjoint copies of C_{3}. In this paper, we determine the anti-Ramsey number ar(n,kC_{3}) for n=3k and n\geq2k^{2}-k+2, respectively. When 3k\leq n\leq 2k^{2}-k+2, we give lower and upper bounds for ar(n, kC_{3}).
The generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We denote by $tF$ the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of $\mathrm{ex}(n,K_s,tK_r)$. We extend this result in three directions. First, we determine $\mathrm{ex}(n,K_s,tK_r)$ exactly for sufficiently large $n$. Second, we determine the asymptotics of the analogous number for $p$-uniform hypergraphs. Third, we determine the order of magnitude of $\mathrm{ex}(n,H,tK_r)$ for every graph $H$, and also …
Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having the most cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $m$ edges and maximum degree at most $r$, the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliques of size $r+1…
Let $Q$ be a finite subgraph of the integer grid $G$ in the plane, and let $T$ be a set of pairs of distinct vertices in $G$, called `terminal pairs'. Escaping a subset $X\subset T\cap Q$ from $Q$ means finding edge disjoint paths from the terminals in $X$ into distinct vertices of a set $L$ in the boundary of $Q$. Here we prove several lemmas for the cases where $Q$ is a $3\times 3$ grid, $L$ is the union of a vertical and horizontal boundary line of $Q$, furthermore, $T$ is a set of four terminal pairs in $G$ such that $|T\cap Q|\geq 5$. These lemmas are applied in [4] and complete the proof…
We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves are labelled by the vertices of a graph $G$, and the width is measured by the maximum possible cut-rank of a partition of $V(G)$ induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth-$2$ rank-brittleness of $G$. We prove that for all $n$, every graph with sufficiently large dept…
Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate this number from above by $|X|^{c(n)}$ where $$ c(n)=\left(1-\frac{(n-1)\ln (n-1)}{n\ln n} \right)^{-1}. $$ This extends the recent result of Kane-Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$, to an arbitrary finite number of disjoint $n-1$ partitions.
A strongly regular graph is called trivial if it or its complement is a union of disjoint cliques. We prove that every infinite family of nontrivial strongly regular graphs is quasi-random in the sense of Chung, Graham and Wilson.
We prove Bowen's conjecture that every sequence of finite graphs that locally converges to the Cayley graph of a countably infinite group with Kazhdan Property (T) is essentially a vertex-disjoint union of expander graphs. We characterize graph sequences that are essentially a vertex-disjoint union of expander graphs in terms of the Markov operator.
Let $G$ be a graph with $m$ edges and let $f$ be a bijection from $E(G)$ to $\{1,2, \dots, m\}$. For any vertex $v$, denote by $ϕ_f(v)$ the sum of $f(e)$ over all edges $e$ incident to $v$. If $ϕ_f(v) \neq ϕ_f(u)$ holds for any two distinct vertices $u$ and $v$, then $f$ is called an {\it antimagic labeling} of $G$. We call $G$ {\it antimagic} if such a labeling exists. Hartsfield and Ringel in 1991 conjectured that all connected graphs except $P_2$ are antimagic. Denote the disjoint union of graphs $G$ and $H$ by $G \cup H$, and the disjoint union of $t$ copies of $G$ by $tG$. For an antimagi…
An \emph{acyclic coloring} of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of \emph{star coloring} requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in O(n) time. We also show that the acyclic chromatic number, the star ch…
We describe two constructions of (very) dense graphs which are edge disjoint unions of large {\em induced} matchings. The first construction exhibits graphs on $N$ vertices with ${N \choose 2}-o(N^2)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o(1)}$. The second construction provides a covering of all edges of the complete graph $K_N$ by two graphs, each being the edge disjoint union of at most $N^{2-δ}$ induced matchings, where $δ> 0.058$. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linia…
A graph $G = (V, E)$ is \emph{partitionable} if there exists a partition $\{A, B\}$ of $V$ such that $A$ induces a disjoint union of cliques and $B$ induces a triangle-free graph. In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be $\NP$-complete on arbitrary graphs. Here it is proved that if a graph $G$ is bull-free, planar, perfect, $K_4$-free or does not contain certain holes then deciding whether $G$ is partitionable is $\NP$-complete. This answers an open question posed by Thomass{é}, Trotignon and Vuškovi{ć}. …
