Search ResultsShowing 1-20 of 108
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arXiv:2501.05049 (Published 2025-01-09)
On the Boxicity of Line Graphs and of Their Complements
Comments: A shorter account of a part of the results of this paper appeared in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization, 20 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2309.02062The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs.'' The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. As every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $NP$-hard: for the existence and optimization of interval-order subgraphs of line graphs, or of interval completions and the boxicity of their complement, if the boxicity is bounded. We finally extend our approach to upper and lower bounding the boxicity of line graphs.
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arXiv:2411.17996 (Published 2024-11-27)
Ramsey--Dirac theory for bounded degree hypertrees
Comments: 25 pages + 4 page appendixCategories: math.CORamsey--Tur\'an theory considers Tur\'an type questions in Ramsey-context, asking for the existence of a small subgraph in a graph $G$ where the complement $\overline{G}$ lacks an appropriate subgraph $F$, such as a clique of linear size. Similarly, one can consider Dirac-type questions in Ramsey context, asking for the existence of a spanning subgraph $H$ in a graph $G$ where the complement $\overline{G}$ lacks an appropriate subgraph $F$, which we call a Ramsey--Dirac theory question. When $H$ is a connected spanning subgraph, the disjoint union $K_{n/2}\cup K_{n/2}$ of two large cliques shows that it is natural to consider complete bipartite graphs $F$. Indeed, Han, Hu, Ping, Wang, Wang and Yang in 2024 proved that if $G$ is an $n$-vertex graph with $\delta(G)=\Omega(n)$ where the complement $\overline{G}$ does not contain any complete bipartite graph $K_{m,m}$ with $m=\Omega(n)$, then $G$ contains every $n$-vertex bounded degree tree $T$ as a subgraph. Extending this result to the Ramsey--Dirac theory for hypertrees, we prove that if $G$ is an $n$-vertex $r$-uniform hypergraph with $\delta(G)=\Omega(n^{r-1})$ where the complement $\overline{G}$ does not contain any complete $r$-partite hypergraph $K_{m,m,\dots, m}$ with $m=\Omega(n)$, then $G$ contains every $n$-vertex bounded degree hypertree $T$ as a subgraph. We also prove the existence of matchings and loose Hamilton cycles in the same setting, which extends the result of Mcdiarmid and Yolov into hypergraphs. This result generalizes the universality result on randomly perturbed graphs by B\"ottcher, Han, Kohayakawa, Montgomery, Parczyk and Person in 2019 into hypergraphs and also strengthen the results on quasirandom hypergraphs by Lenz, Mubayi and Mycroft in 2016 and Lenz and Mubayi in 2016 into hypergraphs satisfying a much weaker pseudorandomness condition.
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arXiv:2408.05949 (Published 2024-08-12)
Strong zero-divisor graph of p.q.-Baer $*$-rings
Categories: math.COIn this paper, we study the strong zero-divisor graph of a p.q.-Baer $*$-ring. We determine the condition on a p.q.-Baer $*$-ring (in terms of the smallest central projection in a lattice of central projections of a $*$-ring), so that its strong zero-divisor graph contains a cut vertex. It is proved that the set of cut vertices of a strong zero-divisor graph of a p.q.-Baer $*$-ring forms a complete subgraph. We prove that the complement of the strong zero-divisor graph of a p.q.-Baer $*$-ring is connected if and only if the $*$-ring contains at least six central projections. We characterize the diameter and girth of the complement of a strong zero-divisor graph of a p.q.-Baer $*$-ring. Also, we characterize p.q.-Baer $*$-rings whose strong zero-divisor graph is complemented.
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arXiv:2407.09476 (Published 2024-07-12)
Intrinsically knotted graphs and connected domination
Comments: 18 pages, 10 figuresWe classify all the maximal linklessly embeddable graphs of order 12 and show that their complements are all intrinsically knotted. We derive results about the connected domination numbers of a graph and its complement. We provide an answer to an open question about the minimal order of a 3-non-compliant graph. We prove that the complements of knotlessly embeddable graphs of order at least 15 are all intrinsically knotted. We provide results on general $k$-non-compliant graphs and leave a set of open questions for further exploration of the subject.
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An improved lower bound on the Shannon capacities of complements of odd cycles
Comments: 7 pages, 1 figureImproving a 2003 result of Bohman and Holzman, we show that for $n \geq 1$, the Shannon capacity of the complement of the $2n+1$-cycle is at least $(2^{r_n} + 1)^{1/r_n} = 2 + \Omega(2^{-r_n}/r_n)$, where $r_n = \exp(O((\log n)^2))$ is the number of partitions of $2(n-1)$ into powers of $2$. We also discuss a connection between this result and work by Day and Johnson in the context of graph Ramsey numbers.
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arXiv:2401.00666 (Published 2024-01-01)
On the $δ$-chromatic numbers of the Cartesian products of graphs
Categories: math.COIn this work, we study the $\delta$-chromatic number of a graph which is the chromatic number of the $\delta$-complement of a graph. We give a structure of the $\delta$-complements and sharp bounds on the $\delta$-chromatic numbers of the Cartesian products of graphs. Furthermore, we compute the $\delta$-chromatic numbers of various classes of Cartesian product graphs, including the Cartesian products between cycles, paths, and stars.
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arXiv:2311.18284 (Published 2023-11-30)
The Complement of the Djokovic-Winkler Relation
The Djokovi\'{c}-Winkler relation $\Theta$ is a binary relation defined on the edge set of a given graph that is based on the distances of certain vertices and which plays a prominent role in graph theory. In this paper, we explore the relatively uncharted ``reflexive complement'' $\overline\Theta$ of $\Theta$, where $(e,f)\in \overline\Theta$ if and only if $e=f$ or $(e,f)\notin \Theta$ for edges $e$ and $f$. We establish the relationship between $\overline\Theta$ and the set $\Delta_{ef}$, comprising the distances between the vertices of $e$ and $f$ and shed some light on the intricacies of its transitive closure $\overline\Theta^*$. Notably, we demonstrate that $\overline\Theta^*$ exhibits multiple equivalence classes only within a restricted subclass of complete multipartite graphs. In addition, we characterize non-trivial relations $R$ that coincide with $\overline\Theta$ as those where the graph representation is disconnected, with each connected component being the (join of) Cartesian product of complete graphs. The latter results imply, somewhat surprisingly, that knowledge about the distances between vertices is not required to determine $\overline\Theta^*$. Moreover, $\overline\Theta^*$ has either exactly one or three equivalence classes.
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arXiv:2308.05442 (Published 2023-08-10)
Optimal chromatic bound for ($P_3\cup P_2$, house)-free graphs
Comments: arXiv admin note: text overlap with arXiv:2307.11946Categories: math.COLet $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup V(H)$ and $E(G\cup H)=E(G)\cup E(H)$. We use $P_k$ to denote a {\em path} on $k$ vertices, use {\em house} to denote the complement of $P_5$. In this paper, we show that $\chi(G)\le2\omega(G)$ if $G$ is ($P_3\cup P_2$, house)-free. Moreover, this bound is optimal when $\omega(G)\ge2$.
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arXiv:2307.09129 (Published 2023-07-18)
Universal adjacency spectrum of (proper) power graphs and their complements on some groups
Categories: math.COThe power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple graph $H$ with adjacency matrix $A(H)$ and degree diagonal matrix $D(H)$, the universal adjacency matrix is $U(H)= \alpha A(H)+\beta D(H)+ \gamma I +\eta J$, where $\alpha (\neq 0), \beta, \gamma, \eta \in \mathbb{R}$, $I$ is the identity matrix and $J$ is the all-ones matrix of suitable order. One can study many graph-associated matrices, such as adjacency, Laplacian, signless Laplacian, Seidel etc. in a unified manner through the universal adjacency matrix of a graph. Here we study universal adjacency eigenvalues and eigenvectors of power graphs, proper power graphs and their complements on the group $\mathbb{Z}_n$, dihedral group ${D}_n$, and the generalized quaternion group ${Q}_n$. Spectral results of no kind for the complement of power graph on any group were obtained before. We determine the full spectrum in some particular cases. Moreover, several existing results can be obtained as very specific cases of some results of the paper.
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arXiv:2305.13630 (Published 2023-05-23)
Near automorphisms of complement of a cycle
Categories: math.COLet $G$ be a graph with vertex set $V(G)$, $f$ a permutation of $V(G)$. Define $\delta_f(G)=|d(x,y)-d(f(x),f(y))|$ and $\delta_f(G)=\Sigma\delta_f(x,y)$, where the sum is taken over all unordered pair $x$, $y$ of distinct vertices of $G$. $\delta_f(x,U)=\Sigma\delta_f(x,y)$, where $U\subseteq V(G)$ and $y\in U$. Let $\pi(G)$ denote the smallest positive value of $\delta_f(G)$ among all permutations of $V(G)$. A permutation $f$ with $\delta_f(G)=\pi(G)$ is called a near automorphisms of $G$. In this paper, the near automorphism of the complement of a cycle is characterized, moreover, $\pi(\overline{C_n})$ is determined.
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arXiv:2301.01708 (Published 2023-01-04)
Extremal problems for the eccentricity matrices of complements of trees
Comments: 18 pages. Preliminary versionCategories: math.COThe eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) $\mathcal{E}$-spectral radius among the complements of trees. Then, we prove that the $\mathcal{E}$-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least $\mathcal{E}$-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest $\mathcal{E}$-eigenvalue and the $\mathcal{E}$-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum type lower bounds for the second largest $\mathcal{E}$-eigenvalue and $\mathcal{E}$-energy of a tree and its complement.
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arXiv:2212.04570 (Published 2022-12-08)
On dominating graph of graphs, median graphs and partial cubes, and graphs in which complement of every minimal dominating set is minimal dominating
Categories: math.COThe dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating graph has become an increasingly interesting subject in domination theory. On the other hand, median graphs and partial cubes are two fundamental graph classes in graph theory. In this paper, we make some new connections between domination theory and the theory of median graphs and partial cubes. As the main result, we show that the following conditions are equivalent for every graph $G \not \simeq C_4$ with no isolated vertex, and in particular, that the simple third condition completely characterizes first two ones in which three concepts of dominating graphs, median graphs and complement of minimal dominating sets get related: - The dominating graph of G is a median graph, - The complement of every minimal dominating set of G is a minimal dominating set, - Every vertex of G is either of degree 1 or adjacent to a vertex of degree 1. As another result, we prove that the dominating graph of every graph is a partial cube and also give some examples to show that not all partial cubes or median graphs are isomorphic to the dominating graph of a graph. The above-mentioned results, as another highlight of the paper, provide novel infinite sources of examples of median graphs and partial cubes.
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arXiv:2211.05033 (Published 2022-11-09)
Rational Homotopy Type Of Complements Of Submanifold Arrangements
We will provide an explicit cdga controlling the rational homotopy type of the complement to a smooth arrangement $X-\cup_i Z_i$ in a smooth compact algebraic variety $X$ over $\mathbb{C}$. This generalizes the corresponding result of Morgan in case of a divisor with normal crossings to arbitrary smooth arrangements. The model is given in terms of the arrangement $Z_i$ and agrees with a model introduced by Chen-L\"u-Wu for computing the cohomology. As an application we reprove a formality theorem due to Feichtner-Yuzvinksy. Then we show that the Kritz-Totaro model computes the rational homotopy type in case of chromatic configuration spaces of smooth compact algebraic varieties.
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Connectedness in Friends-and-Strangers Graphs of Spiders and Complements
Let $X$ and $Y$ be two graphs with vertex set $[n]$. Their friends-and-strangers graph $\mathsf{FS}(X,Y)$ is a graph with vertices corresponding to elements of the group $S_n$, and two permutations $\sigma$ and $\sigma'$ are adjacent if they are separated by a transposition $\{a,b\}$ such that $a$ and $b$ are adjacent in $X$ and $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y$. Specific friends-and-strangers graphs such as $\mathsf{FS}(\mathsf{Path}_n,Y)$ and $\mathsf{FS}(\mathsf{Cycle}_n,Y)$ have been researched, and their connected components have been enumerated using various equivalence relations such as double-flip equivalence. A spider graph is a collection of path graphs that are all connected to a single center point. In this paper, we delve deeper into the question of when $\mathsf{FS}(X,Y)$ is connected when $X$ is a spider and $Y$ is the complement of a spider or a tadpole.
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arXiv:2208.09918 (Published 2022-08-21)
Discrete group actions on 3-manifolds and embeddable Cayley complexes
We prove that a group $\Gamma$ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if $\Gamma$ has a Cayley complex embeddable -- with certain natural restrictions -- in one of the following four 3-manifolds: (i) $\mathbb{S}^3$, (ii) $\mathbb{R}^3$, (iii) $\mathbb{S}^2 \times \mathbb{R}$, (iv) the complement of a tame Cantor set in $\mathbb{S}^3$.
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arXiv:2207.04641 (Published 2022-07-11)
The complement of enhanced power graph of a finite group
Comments: 7 figuresSubjects: 05C25The enhanced power graph $\mathcal{P}_E(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we give an affirmative answer of the question posed by Cameron [6] which states that: Is it true that the complement of the enhanced power graph $\bar{\mathcal{P}_E(G)}$ of a non-cyclic group $G$ has only one connected component apart from isolated vertices? We classify all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G)}$ is bipartite. We show that the graph $\bar{\mathcal{P}_E(G)}$ is weakly perfect. Further, we study the subgraph $\bar{\mathcal{P}_E(G^*)}$ of $\bar{\mathcal{P}_E(G)}$ induced by all the non-isolated vertices of $\bar{\mathcal{P}_E(G)}$. We classify all finite groups $G$ such that the graph is $\bar{\mathcal{P}_E(G^*)}$ is unicyclic and pentacyclic. We prove the non-existence of finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is outerplanar, planar, projective-planar and toroidal, respectively.
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arXiv:2206.03932 (Published 2022-06-08)
The zero forcing number of the complement of a graph
Comments: 20 pages, 11 figuresCategories: math.COMotivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under some conditions, particularly those that do not contain complete bipartite induced subgraphs. We also move well beyond trees and completely study the possible zero forcing numbers for the complements of unicyclic graphs, and examine the zero forcing number for the complements of some specific families of graphs containing more than one cycle.
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arXiv:2205.15189 (Published 2022-05-30)
Independence number of intersection graphs of axis-parallel segments
Comments: 10 pages and 3 figuresWe prove that for any triangle-free intersection graph of $n$ axis-parallel segments in the plane, the independence number $\alpha$ of this graph is at least $\alpha \ge n/4 + \Omega(\sqrt{n})$. We complement this with a construction of a graph in this class satisfying $\alpha \le n/4 + c \sqrt{n}$ for an absolute constant $c$, which demonstrates the optimality of our result.
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arXiv:2205.12437 (Published 2022-05-25)
On the clique behavior and Hellyness of the complements of regular graphs
Categories: math.COA collection of sets is intersecting, if any pair of sets in the collection has nonempty intersection. A collection of sets \(\mathcal{C}\) has the Helly property if any intersecting subcollection has nonempty intersection. A graph is /Helly/ if the collection of maximal complete subgraphs of \(G\) has the Helly property. We prove that if \(G\) is a \(k\)-regular graph with \(n\) vertices such that \(n>3k+\sqrt{2k^{2}-k}\), then the complement \(\bar{G}\) is not Helly. We also consider the problem of whether the properties of Hellyness and convergence under the clique graph operator are equivalent for the complement of \(k\)-regular graphs, for small values of \(k\).
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arXiv:2205.02276 (Published 2022-05-04)
Iterated line graphs with only negative eigenvalues $-2$, their complements and energy
Categories: math.COThe graphs with all equal negative or positive eigenvalues are special kind in the spectral graph theory. In this article, several iterated line graphs $\mathcal{L}^k(G)$ with all equal negative eigenvalues $-2$ are characterized for $k\ge 1$ and their energy consequences are presented. Also, the spectra and the energy of complement of these graphs are obtained, interestingly they have exactly two positive eigenvalues with different multiplicities. Moreover, we characterize a large class of equienergetic graphs which generalize some of the existing results. There are two different quotient matrices defined for an equitable partition of $H$-join (generalized composition) of regular graphs to find the spectrum (partial) of adjacency matrix, Laplacian matrix and signless Laplacian matrix, it has been proved that these two quotient matrices give the same respective spectrum of graphs.