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The Existence of a Path with Two Blocks in Digraphs Graphs Comb. (IF 0.7) Pub Date : 2024-03-12 Amine El Sahili, Maidoun Mortada, Sara Nasser
We give a new elementary proof of El Sahili conjecture El Sahili (Discrete Math 287:151–153, 2004) stating that any n-chromatic digraph D, with \(n\ge 4\), contains a path with two blocks of order n.
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A Note on the Gyárfás–Sumner Conjecture Graphs Comb. (IF 0.7) Pub Date : 2024-03-08 Tung Nguyen, Alex Scott, Paul Seymour
The Gyárfás–Sumner conjecture says that for every tree T and every integer \(t\ge 1\), if G is a graph with no clique of size t and with sufficiently large chromatic number, then G contains an induced subgraph isomorphic to T. This remains open, but we prove that under the same hypotheses, G contains a subgraph H isomorphic to T that is “path-induced”; that is, for some distinguished vertex r, every
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The Spectral Radius, Maximum Average Degree and Cycles of Consecutive Lengths of Graphs Graphs Comb. (IF 0.7) Pub Date : 2024-03-05 Wenqian Zhang
In this paper, we study the relationship between spectral radius and maximum average degree of graphs. By using this relationship and the previous technique of Li and Ning in (J Graph Theory 103:486–492, 2023), we prove that, for any given positive number \(\varepsilon <\frac{1}{3}\), if n is a sufficiently large integer, then any graph G of order n with \(\rho (G)>\sqrt{\left\lfloor \frac{n^{2}}{4}\right\rfloor
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On Restricted Intersections and the Sunflower Problem Graphs Comb. (IF 0.7) Pub Date : 2024-03-04 Jeremy Chizewer
A sunflower with r petals is a collection of r sets over a ground set X such that every element in X is in no set, every set, or exactly one set. Erdős and Rado [5] showed that a family of sets of size n contains a sunflower if there are more than \(n!(r-1)^n\) sets in the family. Alweiss et al. [1] and subsequently, Rao [7] and Bell et al. [2] improved this bound to \((O(r \log n))^n\). We study the
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Infinite Families of k-Vertex-Critical ( $$P_5$$ , $$C_5$$ )-Free Graphs Graphs Comb. (IF 0.7) Pub Date : 2024-02-25 Ben Cameron, Chính Hoàng
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Planar Turán Numbers of Cubic Graphs and Disjoint Union of Cycles Graphs Comb. (IF 0.7) Pub Date : 2024-02-25
Abstract The planar Turán number of a graph H, denoted by \(ex_{_\mathcal {P}}(n,H)\) , 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
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Weak Dynamic Coloring of Planar Graphs Graphs Comb. (IF 0.7) Pub Date : 2024-02-24 Caroline Accurso, Vitaliy Chernyshov, Leaha Hand, Sogol Jahanbekam, Paul Wenger
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Borodin–Kostochka Conjecture Holds for Odd-Hole-Free Graphs Graphs Comb. (IF 0.7) Pub Date : 2024-02-10 Rong Chen, Kaiyang Lan, Xinheng Lin, Yidong Zhou
The Borodin–Kostochka Conjecture states that for a graph G, if \(\Delta (G)\ge 9\), then \(\chi (G)\le \max \{\Delta (G)-1,\omega (G)\}\). In this paper, we prove the Borodin–Kostochka Conjecture holding for odd-hole-free graphs.
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Parallel Connectivity in Edge-Colored Complete Graphs: Complexity Results Graphs Comb. (IF 0.7) Pub Date : 2024-02-10 Rachid Saad
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On Inducing Degenerate Sums Through 2-Labellings Graphs Comb. (IF 0.7) Pub Date : 2024-02-09 Julien Bensmail, Hervé Hocquard, Pierre-Marie Marcille
We deal with a variant of the 1–2–3 Conjecture introduced by Gao, Wang, and Wu (Graphs Combin 32:1415–1421, 2016) . This variant asks whether all graphs can have their edges labelled with 1 and 2 so that when computing the sums of labels incident to the vertices, no monochromatic cycle appears. In the aforementioned seminal work, the authors mainly verified their conjecture for a few classes of graphs
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Infinite Ramsey-Minimal Graphs for Star Forests Graphs Comb. (IF 0.7) Pub Date : 2024-02-09 Fawwaz Fakhrurrozi Hadiputra, Valentino Vito
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Multipermutations and Stirling Multipermutations Graphs Comb. (IF 0.7) Pub Date : 2024-02-07 Richard A. Brualdi, Geir Dahl
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Lattice Path Bicircular Matroids Graphs Comb. (IF 0.7) Pub Date : 2024-02-07
Abstract Lattice path matroids and bicircular matroids are two well-known classes of transversal matroids. In the seminal work of Bonin and de Mier about structural properties of lattice path matroids, the authors claimed that lattice path matroids significantly differ from bicircular matroids. Recently, it was proved that all cosimple lattice path matroids have positive double circuits, while it was
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The Strong Spectral Property of Graphs: Graph Operations and Barbell Partitions Graphs Comb. (IF 0.7) Pub Date : 2024-02-06 Sarah Allred, Emelie Curl, Shaun Fallat, Shahla Nasserasr, Houston Schuerger, Ralihe R. Villagrán, Prateek K. Vishwakarma
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A Novel Count of the Spanning Trees of a Cube Graphs Comb. (IF 0.7) Pub Date : 2024-01-28 Thomas W. Mattman
Using the special value at \(u=1\) of the Artin-Ihara L-function, we give a short proof of the count of the number of spanning trees in the n-cube.
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The Oriented Diameter of Graphs with Given Connected Domination Number and Distance Domination Number Graphs Comb. (IF 0.7) Pub Date : 2024-01-28 Peter Dankelmann, Jane Morgan, Emily Rivett-Carnac
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Crossing and intersecting families of geometric graphs on point sets Graphs Comb. (IF 0.7) Pub Date : 2024-01-25 J. L. Álvarez-Rebollar, J. Cravioto-Lagos, N. Marín, O. Solé-Pi, J. Urrutia
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A Construction of Optimal 1-Spontaneous Emission Error Designs Graphs Comb. (IF 0.7) Pub Date : 2024-01-19 Junling Zhou, Na Zhang
A t-spontaneous emission error design, denoted by t-(v, k; m) SEED or t-SEED in short, is a system \({{\mathcal {B}}}\) of k-subsets of a v-set V with a partition \({{\mathcal {B}}}_1,\mathcal{B}_2,\ldots ,{{\mathcal {B}}}_{m}\) of \({{\mathcal {B}}}\) satisfying \({{|\{B\in {\mathcal {B}}_i:\, E \subseteq B\}|}\over {|{\mathcal {B}}_i|}}=\mu _E \) for any \(1\le i\le m\) and \(E\subseteq V\), \(|E|\le
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Path Planning in a Weighted Planar Subdivision Under the Manhattan Metric Graphs Comb. (IF 0.7) Pub Date : 2024-01-19 Mansoor Davoodi, Ashkan Safari
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Compatible Spanning Circuits and Forbidden Induced Subgraphs Graphs Comb. (IF 0.7) Pub Date : 2024-01-19 Zhiwei Guo, Christoph Brause, Maximilian Geißer, Ingo Schiermeyer
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Fixing Numbers of Graphs with Symmetric and Generalized Quaternion Symmetry Groups Graphs Comb. (IF 0.7) Pub Date : 2024-01-19 Christina Graves, L.-K. Lauderdale
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Equality of Ordinary and Symbolic Powers of Some Classes of Monomial Ideals Graphs Comb. (IF 0.7) Pub Date : 2024-01-19 Kanoy Kumar Das
In this article, our aim is to extend the class of monomial ideals for which symbolic and ordinary powers coincide. This property has been characterized for the class of edge ideals of simple graphs, and in this article, we study a completely new class of monomial ideals associated to simple graphs, namely the class of generalized edge ideals. We give a complete description of the primary components
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Coloring of Graphs Avoiding Bicolored Paths of a Fixed Length Graphs Comb. (IF 0.7) Pub Date : 2024-01-11 Alaittin Kırtışoğlu, Lale Özkahya
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Signed Ramsey Numbers Graphs Comb. (IF 0.7) Pub Date : 2023-12-28 Mohammed A. Mutar, Vaidy Sivaraman, Daniel Slilaty
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Erdős–Hajnal Problem for H-Free Hypergraphs Graphs Comb. (IF 0.7) Pub Date : 2023-12-28 Danila Cherkashin, Alexey Gordeev, Georgii Strukov
This paper deals with the minimum number \(m_H(r)\) of edges in an H-free hypergraph with the chromatic number more than r. We show how bounds on Ramsey and Turán numbers imply bounds on \(m_H(r)\).
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Four-vertex traces of finite sets Graphs Comb. (IF 0.7) Pub Date : 2023-12-23 Peter Frankl, Jian Wang
Let \([n]=X_1\cup X_2\cup X_3\) be a partition with \(\lfloor \frac{n}{3}\rfloor \le |X_i|\le \lceil \frac{n}{3}\rceil \) and define \({\mathcal {G}}=\{G\subset [n]:|G\cap X_i|\le 1, 1\le i\le 3\}\). It is easy to check that the trace \({\mathcal {G}}_{\mid Y}:=\{G\cap Y:G\in {\mathcal {G}}\}\) satisfies \(|{\mathcal {G}}_{\mid Y}|\le 12\) for all 4-sets \(Y\subset [n]\). In the present paper, we prove
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Weak-Dynamic Coloring of Graphs Beyond-Planarity Graphs Comb. (IF 0.7) Pub Date : 2023-12-23 Weichan Liu, Guiying Yan
A weak-dynamic coloring of a graph is a vertex coloring (not necessarily proper) in such a way that each vertex of degree at least two sees at least two colors in its neighborhood. It is proved that the weak-dynamic chromatic number of the class of k-planar graphs (resp. IC-planar graphs) is equal to (resp. at most) the chromatic number of the class of 2k-planar graphs (resp. 1-planar graphs), and
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Some New Constructions of Difference Systems of Sets Graphs Comb. (IF 0.7) Pub Date : 2023-12-23 Shuyu Shen, Jingjun Bao
Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein, which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we describe four direct constructions of optimal DSSs from finite projective geometries and present a recursive construction of DSSs by extending the known construction. As a consequence, new
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The Maximum 4-Vertex-Path Packing of a Cubic Graph Covers At Least Two-Thirds of Its Vertices Graphs Comb. (IF 0.7) Pub Date : 2023-12-20 Wenying Xi, Wensong Lin
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Tashkinov-Trees: An Annotated Proof Graphs Comb. (IF 0.7) Pub Date : 2023-12-14 András Sebő
Tashkinov-trees have been used as a tool for proving bounds on the chromatic index, and are becoming a fundamental tool for edge-coloring. Was its publication in a language different from English an obstacle for the accessibility of a clean and complete proof of Tashkinov’s fundamental theorem? Tashkinov’s original, Russian paper offers a clear presentation of this theorem and its proof. The theorem
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Turán Numbers of Several Bipartite Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-12-14 Ye Wang, Yusheng Li, Yan Li
For graphs \(H_1,H_2,\dots ,H_k\), the k-color Turán number \(ex(n,H_1,H_2,\dots ,H_k)\) is the maximum number of edges in a k-colored graph of order n that does not contain monochromatic \(H_i\) in color i as a subgraph, where \(1\le i\le k\). In this note, we show that if \(H_i\) is a bipartite graph with at least two edges for \(1\le i\le k\), then \(ex(n,H_1,H_2,\dots ,H_k)=(1+o(1))\sum _{i=1}^kex(n
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On the Turán Number of $$K_m \vee C_{2k-1}$$ Graphs Comb. (IF 0.7) Pub Date : 2023-12-04 Jingru Yan
Given a graph H and a positive integer n, the Turán number of H of the order n, denoted by ex(n, H), is the maximum size of a simple graph of order n that does not contain H as a subgraph. Given graphs G and H, \(G \vee H\) denotes the join of G and H. In this paper, we prove \(ex(n, K_m \vee C_{2k-1}) = \left\lfloor \frac{(m+1)n^2}{2(m+2)}\right\rfloor \) for \(n\ge 2(m+2)k-3(m+2)-1\).
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Complexity of Total Dominator Coloring in Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-11-28 Michael A. Henning, Kusum, Arti Pandey, Kaustav Paul
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$$l_{1}$$ -embeddability of shifted quadrilateral cylinder graphs Graphs Comb. (IF 0.7) Pub Date : 2023-11-29 Guangfu Wang, Zhikun Xiong, Lijun Chen
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Size Gallai–Ramsey Number Graphs Comb. (IF 0.7) Pub Date : 2023-11-23 Yaping Mao
The size Ramsey number \({\hat{\textrm{r}}}(G,H)\) of two graphs G, H, introduced by Erdös et al., is defined as \({\hat{\textrm{r}}}(G,H)=\min \{|E(F)|:F\longrightarrow (G,H)\}\). Recently, Gallai–Ramsey number \(\textrm{gr}_k\) has been studied a lot. In this paper, we introduce the concept of size Gallai–Ramsey number (Simply, SGR number) of graphs. Given two graphs G and H, the size Gallai–Ramsey
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The Chromatic Number of a Graph with Two Odd Holes and an Odd Girth Graphs Comb. (IF 0.7) Pub Date : 2023-11-23 Kaiyang Lan, Feng Liu
An odd hole is an induced odd cycle of length at least five. Let \(\ell \ge 2\) be an integer, and let \({\mathcal {G}}_\ell \) denote the family of graphs which have girth \(2\ell + 1\) and have no holes of odd length at least \(2\ell +5\). In this paper, we prove that every graph \(G \in \cup _{\ell \ge 3}{\mathcal {G}}_\ell \) is 4-colourable.
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Proper Cycles and Rainbow Cycles in 2-triangle-free edge-colored Complete Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-11-20 Shanshan Guo, Fei Huang, Jinjiang Yuan
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Mutually Orthogonal Sudoku Latin Squares and Their Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-11-14 Sho Kubota, Sho Suda, Akane Urano
We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares.
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Toric Rings of Perfectly Matchable Subgraph Polytopes Graphs Comb. (IF 0.7) Pub Date : 2023-11-04 Kenta Mori
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Cubic Graphs Admitting Vertex-Transitive Almost Simple Groups Graphs Comb. (IF 0.7) Pub Date : 2023-10-28 Jia-Li Du, Fu-Gang Yin, Menglin Ding
Let \({\varGamma }\) be a connected cubic graph admitting a vertex-transitive almost simple group G of automorphisms. In this paper, we study the normality of the socle T of G in the full automorphism group \(\text {Aut}({\varGamma })\) of \({\varGamma } \). It is proved that if T is not normal in \(\text {Aut}({\varGamma })\), then \(T= \text {A}_{47}\), \(\text {A}_{23}\), \(\text {A}_{2^f-1}\) with
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On Quasi-strongly Regular Graphs with Parameters $$(n,k,a;k-2,c_{2})$$ (I) Graphs Comb. (IF 0.7) Pub Date : 2023-10-17 Jiayi Xie, Gengsheng Zhang
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Every planar graph without 4-cycles and 5-cycles is (3,3)-colorable Graphs Comb. (IF 0.7) Pub Date : 2023-10-17 Xiangwen Li, Jie Liu, Jian-Bo Lv
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A Sharp Upper Bound on the Cycle Isolation Number of Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-10-14 Qing Cui, Jingshu Zhang
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Strong List-Chromatic Index of Planar Graphs with Ore-Degree at Most Seven Graphs Comb. (IF 0.7) Pub Date : 2023-10-09 Mingfang Huang, Guorong Liu, Xiangqian Zhou
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A (2, 1)-Decomposition of Planar Graphs Without Intersecting 3-Cycles and Adjacent $$4^-$$ -Cycles Graphs Comb. (IF 0.7) Pub Date : 2023-10-06 Fangyu Tian, Xiangwen Li
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On the Roots of (Signless) Laplacian Permanental Polynomials of Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-10-06 Tingzeng Wu, Xiaolin Zeng, Huazhong Lü
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Maximum Rectilinear Crossing Number of Uniform Hypergraphs Graphs Comb. (IF 0.7) Pub Date : 2023-10-06 Rahul Gangopadhyay, Ayan
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Modified Erdős–Ginzburg–Ziv Constants for $$\mathbb {Z}_2^{d}$$ Graphs Comb. (IF 0.7) Pub Date : 2023-09-26 Alexander Sidorenko
Let G be a finite abelian group written additively, and let r be a multiple of its exponent. The modified Erdős–Ginzburg–Ziv constant \(\textsf{s}_r'(G)\) is the smallest integer s such that every zero-sum sequence of length at least s over G has a zero-sum subsequence of length r. We find exact values of \(\textsf{s}_{2k}'(\mathbb {Z}_2^d)\) for \(d \le 2k+1\).
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The $$\textbf{uvu}$$ -Avoiding (a, b, c)-Generalized Motzkin Paths with Vertical Steps: Bijections and Statistic Enumerations Graphs Comb. (IF 0.7) Pub Date : 2023-09-14 Yidong Sun, Weichen Wang, Cheng Sun
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Minimizing Visible Edges in Polyhedra Graphs Comb. (IF 0.7) Pub Date : 2023-09-14 Csaba D. Tóth, Jorge Urrutia, Giovanni Viglietta
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Frames and Doubly Resolvable Group Divisible Designs with Block Size Three and Index Two Graphs Comb. (IF 0.7) Pub Date : 2023-08-27 Xiaoyuan Dong, Jinhua Wang
In this paper, we use the intransitive starter-adder method and the standard starter-adder method to construct some new frames and doubly resolvable group divisible designs. Some infinite classes of frames and doubly resolvable group divisible designs are obtained by recursive constructions. On this basis, we almost establish the existence of frames and doubly resolvable group divisible designs with
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Weighted Tutte–Grothendieck Polynomials of Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-08-22 Himadri Shekhar Chakraborty, Tsuyoshi Miezaki, Chong Zheng
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The Global Resilience of Hamiltonicity in G(n, p) Graphs Comb. (IF 0.7) Pub Date : 2023-08-22 Yahav Alon
Denote by \(r_g(G,{\mathscr {H}})\) the global resilience of a graph G with respect to Hamiltonicity. That is, \(r_g(G,{\mathscr {H}})\) is the minimal r for which there exists a subgraph \(H\subseteq G\) with r edges, such that \(G\setminus H\) is not Hamiltonian. We show that if p is above the Hamiltonicity threshold and \(G\sim G(n,p)\) then, with high probability, (We say that a sequence of events
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Injectively $$k$$ -Colored Rooted Forests Graphs Comb. (IF 0.7) Pub Date : 2023-08-22 Thomas Einolf, Robert Muth, Jeffrey Wilkinson
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On Dominating Graph of Graphs, Median Graphs, Partial Cubes and Complement of Minimal Dominating Sets Graphs Comb. (IF 0.7) Pub Date : 2023-08-22 Alireza Mofidi
The 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
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A Note on $$\Delta $$ -Critical Graphs Graphs Comb. (IF 0.7) Pub Date : 2023-08-18 Penny Haxell, Reza Naserasr