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A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs J. Graph Theory (IF 0.9) Pub Date : 2024-04-23 Yan‐Ting Xie, Yong‐De Feng, Shou‐Jun Xu
Partial cubes are the graphs which can be embedded into hypercubes. The cube polynomial of a graph is a counting polynomial of induced hypercubes of , which is defined as , where is the number of induced ‐cubes (hypercubes of dimension ) of . The clique polynomial of is defined as , where () is the number of ‐cliques in and . Equivalently, is exactly the independence polynomial of the complement of
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Flexible list colorings: Maximizing the number of requests satisfied J. Graph Theory (IF 0.9) Pub Date : 2024-04-18 Hemanshu Kaul, Rogers Mathew, Jeffrey A. Mudrock, Michael J. Pelsmajer
Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose , is a graph, is a list assignment for , and is a function with nonempty domain such that for each ( is called a request of ). The triple is ‐satisfiable if there exists a proper ‐coloring of such that for at least vertices in . We say is ‐flexible if is ‐satisfiable whenever is a ‐assignment for and is a request of
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Ramsey numbers for multiple copies of sparse graphs J. Graph Theory (IF 0.9) Pub Date : 2024-04-12 Aurelio Sulser, Miloš Trujić
For a graph and an integer , we let denote the disjoint union of copies of . In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for , one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant such that , provided is sufficiently large. Subsequently, Burr gave an implicit way of computing and noted that this long‐term behaviour occurs
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Bisimplicial separators J. Graph Theory (IF 0.9) Pub Date : 2024-04-12 Martin Milanič, Irena Penev, Nevena Pivač, Kristina Vušković
A minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use
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Best possible upper bounds on the restrained domination number of cubic graphs J. Graph Theory (IF 0.9) Pub Date : 2024-04-12 Boštjan Brešar, Michael A. Henning
A dominating set in a graph is a set of vertices such that every vertex in is adjacent to a vertex in . A restrained dominating set of is a dominating set with the additional restraint that the graph obtained by removing all vertices in is isolate‐free. The domination number and the restrained domination number are the minimum cardinalities of a dominating set and restrained dominating set, respectively
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Graphs with girth 9 and without longer odd holes are 3‐colourable J. Graph Theory (IF 0.9) Pub Date : 2024-04-12 Yan Wang, Rong Wu
For a number , let denote the family of graphs which have girth and have no odd hole with length greater than . Wu, Xu and Xu conjectured that every graph in is 3‐colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in , and is 3‐colourable, respectively. In this paper, we prove that every graph in is 3‐colourable. This confirms Wu, Xu and Xu's conjecture.
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Edge‐minimum saturated k $k$‐planar drawings J. Graph Theory (IF 0.9) Pub Date : 2024-03-29 Steven Chaplick, Fabian Klute, Irene Parada, Jonathan Rollin, Torsten Ueckerdt
For a class of drawings of loopless (multi‐)graphs in the plane, a drawing is saturated when the addition of any edge to results in —this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on ‐planar drawings, that is, graphs drawn in the plane where each edge is crossed at most times, and the classes of all ‐planar drawings obeying a number
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Circular flows in mono‐directed signed graphs J. Graph Theory (IF 0.9) Pub Date : 2024-03-20 Jiaao Li, Reza Naserasr, Zhouningxin Wang, Xuding Zhu
In this paper, the concept of circular ‐flow in a mono‐directed signed graph is introduced. That is a pair , where is an orientation on and satisfies that for each positive edge and for each negative edge , and the total in‐flow equals the total out‐flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi‐directed graphs
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The chromatic number of heptagraphs J. Graph Theory (IF 0.9) Pub Date : 2024-03-20 Di Wu, Baogang Xu, Yian Xu
A pentagraph is a graph without cycles of length 3 or 4 and without induced cycles of odd length at least 7, and a heptagraph is one without cycles of length less than 7 and without induced cycles of odd length at least 9. Chudnovsky and Seymour proved that every pentagraph is 3‐colorable. In this paper, we show that every heptagraph is 3‐colorable.
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Ramsey numbers upon vertex deletion J. Graph Theory (IF 0.9) Pub Date : 2024-03-19 Yuval Wigderson
Given a graph , its Ramsey number is the minimum so that every two‐coloring of contains a monochromatic copy of . It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from , the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number
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Flip distance and triangulations of a polyhedron J. Graph Theory (IF 0.9) Pub Date : 2024-03-19 Zili Wang
It is known that the flip distance between two triangulations of a convex polygon is related to the smallest number of tetrahedra in the triangulation of some polyhedron. The latter was used to compute the diameter of the flip graph of convex polygons with a large number of vertices. However, it is yet unknown whether the flip distance and this smallest number of tetrahedra are always the same or even
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Finding triangle‐free 2‐factors in general graphs J. Graph Theory (IF 0.9) Pub Date : 2024-03-08 David Hartvigsen
A 2‐factor in a graph is a subset of edges such that every node of is incident with exactly two edges of . Many results are known concerning 2‐factors including a polynomial‐time algorithm for finding 2‐factors and a characterization of those graphs that have a 2‐factor. The problem of finding a 2‐factor in a graph is a relaxation of the NP‐hard problem of finding a Hamilton cycle. A stronger relaxation
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Orientations of graphs with maximum Wiener index J. Graph Theory (IF 0.9) Pub Date : 2024-03-04 Zhenzhen Li, Baoyindureng Wu
In this paper, we study the Wiener index of the orientation of trees and theta‐graphs. An orientation of a tree is called no‐zig‐zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree achieving the maximum Wiener index is no‐zig‐zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski
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Sharp lower bounds for the number of maximum matchings in bipartite multigraphs J. Graph Theory (IF 0.9) Pub Date : 2024-03-04 Alexandr V. Kostochka, Douglas B. West, Zimu Xiang
We study the minimum number of maximum matchings in a bipartite multigraph with parts and under various conditions, refining the well‐known lower bound due to M. Hall. When , every vertex in has degree at least , and every vertex in has at least distinct neighbors, the minimum is when and is when . When every vertex has at least two neighbors and , the minimum is , where . We also determine the minimum
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Extremal spectral results of planar graphs without vertex‐disjoint cycles J. Graph Theory (IF 0.9) Pub Date : 2024-02-29 Longfei Fang, Huiqiu Lin, Yongtang Shi
Given a planar graph family , let and be the maximum size and maximum spectral radius over all ‐vertex ‐free planar graphs, respectively. Let be the disjoint union of copies of ‐cycles, and be the family of vertex‐disjoint cycles without length restriction. Tait and Tobin determined that is the extremal spectral graph among all planar graphs with sufficiently large order , which implies the extremal
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Self-avoiding walks and polygons on hyperbolic graphs J. Graph Theory (IF 0.9) Pub Date : 2024-02-19 Christoforos Panagiotis
We prove that for the d$d$-regular tessellations of the hyperbolic plane by k$k$-gons, there are exponentially more self-avoiding walks of length n$n$ than there are self-avoiding polygons of length n$n$. We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed k$k$, we show that the connective constant
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Threshold for stability of weak saturation J. Graph Theory (IF 0.9) Pub Date : 2024-02-23 Mohammadreza Bidgoli, Ali Mohammadian, Behruz Tayfeh‐Rezaie, Maksim Zhukovskii
We study the weak ‐saturation number of the Erdős–Rényi random graph , denoted by , where is the complete graph on vertices. In 2017, Korándi and Sudakov proved that the weak ‐saturation number of is stable, in the sense that it remains the same after removing edges with constant probability. In this paper, we prove that there exists a threshold for this stability property and give upper and lower
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Polyhedra without cubic vertices are prism-hamiltonian J. Graph Theory (IF 0.9) Pub Date : 2024-02-19 Simon Špacapan
The prism over a graph G$G$ is the Cartesian product of G$G$ with the complete graph on two vertices. A graph G$G$ is prism-hamiltonian if the prism over G$G$ is hamiltonian. We prove that every polyhedral graph (i.e., 3-connected planar graph) of minimum degree at least four is prism-hamiltonian.
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The structure of digraphs with excess one J. Graph Theory (IF 0.9) Pub Date : 2024-02-19 James Tuite
A digraph is ‐geodetic if for any (not necessarily distinct) vertices there is at most one directed walk from to with length not exceeding . The order of a ‐geodetic digraph with minimum out‐degree is bounded below by the directed Moore bound . The Moore bound can be met only in the trivial cases and , so it is of interest to look for ‐geodetic digraphs with out‐degree and smallest possible order
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Degree criteria and stability for independent transversals J. Graph Theory (IF 0.9) Pub Date : 2024-02-14 Penny Haxell, Ronen Wdowinski
An independent transversal (IT) in a graph G$G$ with a given vertex partition P${\mathscr{P}}$ is an independent set of vertices of G$G$ (i.e., it induces no edges), that consists of one vertex from each part (block) of P${\mathscr{P}}$. Over the years, various criteria have been established that guarantee the existence of an IT, often given in terms of P${\mathscr{P}}$ being t$t$-thick, meaning all
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Inclusion chromatic index of random graphs J. Graph Theory (IF 0.9) Pub Date : 2024-02-14 Jakub Kwaśny, Jakub Przybyło
Erdős and Wilson proved in 1977 that almost all graphs have chromatic index equal to their maximum degree. In 2001 Balister extended this result and proved that the same number of colours is almost always sufficient if we additionally demand the distinctness of the sets of colours incident with any two vertices. We study a stronger condition and show that one more colour is almost always sufficient
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The overfull conjecture on graphs of odd order and large minimum degree J. Graph Theory (IF 0.9) Pub Date : 2024-02-14 Songling Shan
Let G$G$ be a simple graph with maximum degree Δ(G)${\rm{\Delta }}(G)$. A subgraph H$H$ of G$G$ is overfull if ∣E(H)∣>Δ(G)⌊12∣V(H)∣⌋$| E(H)| \gt {\rm{\Delta }}(G)\lfloor \frac{1}{2}| V(H)| \rfloor $. Chetwynd and Hilton in 1986 conjectured that a graph G$G$ with Δ(G)>13∣V(G)∣${\rm{\Delta }}(G)\gt \frac{1}{3}| V(G)| $ has chromatic index Δ(G)${\rm{\Delta }}(G)$ if and only if G$G$ contains
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On tree decompositions whose trees are minors J. Graph Theory (IF 0.9) Pub Date : 2024-02-11 Pablo Blanco, Linda Cook, Meike Hatzel, Claire Hilaire, Freddie Illingworth, Rose McCarty
In 2019, Dvořák asked whether every connected graph G$G$ has a tree decomposition (T,B)$(T,{\rm{ {\mathcal B} }})$ so that T$T$ is a subgraph of G$G$ and the width of (T,B)$(T,{\rm{ {\mathcal B} }})$ is bounded by a function of the treewidth of G$G$. We prove that this is false, even when G$G$ has treewidth 2 and T$T$ is allowed to be a minor of G$G$.
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Minimum degree stability of C 2 k + 1-free graphs J. Graph Theory (IF 0.9) Pub Date : 2024-02-11 Xiaoli Yuan, Yuejian Peng
We consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a C2k+1${C}_{2k+1}$-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a {C3,C5,…,C2k+1}$\{{C}_{3},{C}_{5},\ldots ,{C}_{2k+1}\}$-free graph on n$n$ vertices has minimum degree
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Integer flows on triangularly connected signed graphs J. Graph Theory (IF 0.9) Pub Date : 2024-02-08 Liangchen Li, Chong Li, Rong Luo, Cun-Quan Zhang
A triangle-path in a graph G$G$ is a sequence of distinct triangles T1,T2,…,Tm${T}_{1},{T}_{2},\ldots ,{T}_{m}$ in G$G$ such that for any i,j$i,j$ with 1≤ii+1$j\gt i+1$. A connected graph G$G$ is triangularly connected if for any two nonparallel edges e$e$ and e′$e^{\prime} $ there is a triangle-path T1T2⋯Tm${T}_{1}{T}_{2}\cdots {T}_{m}$ such that e∈E(T1)$e\in E({T}_{1})$ and e′∈E(Tm)$e^{\prime}
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A lower bound for the complex flow number of a graph: A geometric approach J. Graph Theory (IF 0.9) Pub Date : 2024-02-04 Davide Mattiolo, Giuseppe Mazzuoccolo, Jozef Rajník, Gloria Tabarelli
Let r≥2$r\ge 2$ be a real number. A complex nowhere-zero r$r$-flow on a graph G$G$ is an orientation of G$G$ together with an assignment φ:E(G)→C$\varphi :E(G)\to {\mathbb{C}}$ such that, for all e∈E(G)$e\in E(G)$, the Euclidean norm of the complex number φ(e)$\varphi (e)$ lies in the interval [1,r−1]$[1,r-1]$ and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow
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On two cycles of consecutive even lengths J. Graph Theory (IF 0.9) Pub Date : 2024-01-24 Jun Gao, Binlong Li, Jie Ma, Tianying Xie
Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two. We prove the following average degree counterpart that every n$n$-vertex graph G$G$ with at least 52(n−1)$\frac{5}{2}(n-1)$ edges, unless 4|(n−1)$4|(n-1)$ and every block of G$G$ is a clique K5${K}_{5}$, contains two cycles of consecutive even lengths. Our proof is mainly
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A proof of Frankl–Kupavskii's conjecture on edge-union condition J. Graph Theory (IF 0.9) Pub Date : 2024-01-07 Hongliang Lu, Xuechun Zhang
A 3-graph F${\rm{ {\mathcal F} }}$ is U(s,2s+1)$U(s,2s+1)$ if for any s$s$ edges e1,…,es∈E(F)${e}_{1},\ldots ,{e}_{s}\in E({\rm{ {\mathcal F} }})$, ∣e1∪⋯∪es∣≤2s+1$| {e}_{1}\cup \cdots \cup {e}_{s}| \le 2s+1$. Frankl and Kupavskii proposed the following conjecture: For any 3-graph F${\rm{ {\mathcal F} }}$ with n$n$ vertices, if F${\rm{ {\mathcal F} }}$ is U(s,2s+1)$U(s,2s+1)$, then
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Arc-disjoint out-branchings and in-branchings in semicomplete digraphs J. Graph Theory (IF 0.9) Pub Date : 2024-01-03 J. Bang-Jensen, Y. Wang
An out-branching Bu+${B}_{u}^{+}$ (in-branching Bu−${B}_{u}^{-}$) in a digraph D$D$ is a connected spanning subdigraph of D$D$ in which every vertex except the vertex u$u$, called the root, has in-degree (out-degree) one. It is well known that there exists a polynomial algorithm for deciding whether a given digraph has k$k$ arc-disjoint out-branchings with prescribed roots (k$k$ is part of the input)
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New eigenvalue bound for the fractional chromatic number J. Graph Theory (IF 0.9) Pub Date : 2023-12-27 Krystal Guo, Sam Spiro
Given a graph G $G$ , we let s + ( G ) ${s}^{+}(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of G $G$ , and we similarly define s − ( G ) ${s}^{-}(G)$ . We prove that
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Optimal linear-Vizing relationships for (total) domination in graphs J. Graph Theory (IF 0.9) Pub Date : 2023-12-18 Michael A. Henning, Paul Horn
A total dominating set in a graph G$G$ is a set of vertices of G$G$ such that every vertex is adjacent to a vertex of the set. The total domination number γt(G)${\gamma }_{t}(G)$ is the minimum cardinality of a total dominating set in G$G$. In this paper, we study the following open problem posed by Yeo. For each Δ≥3${\rm{\Delta }}\ge 3$, find the smallest value, rΔ${r}_{{\rm{\Delta }}}$, such that
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Classes of intersection digraphs with good algorithmic properties J. Graph Theory (IF 0.9) Pub Date : 2023-12-18 Lars Jaffke, O-joung Kwon, Jan Arne Telle
While intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection digraphs in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied
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Exact values for some unbalanced Zarankiewicz numbers J. Graph Theory (IF 0.9) Pub Date : 2023-12-14 Guangzhou Chen, Daniel Horsley, Adam Mammoliti
For positive integers s $s$ , t $t$ , m $m$ and n $n$ , the Zarankiewicz number Z s , t ( m , n ) ${Z}_{s,t}(m,n)$ is defined to be the maximum number of edges in a bipartite graph with parts of sizes m $m$ and n $n$ that has no complete bipartite subgraph containing s $s$ vertices in the part of size m $m$ and t $t$ vertices in the part of size n $n$ . A simple argument shows that, for each t ≥ 2
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Overfullness of edge-critical graphs with small minimal core degree J. Graph Theory (IF 0.9) Pub Date : 2023-12-13 Yan Cao, Guantao Chen, Guangming Jing, Songling Shan
Let G$G$ be a simple graph. Let Δ(G)${\rm{\Delta }}(G)$ and χ′(G)$\chi ^{\prime} (G)$ be the maximum degree and the chromatic index of G$G$, respectively. We call G$G$ overfull if ∣E(G)∣∕⌊∣V(G)∣∕2⌋>Δ(G)$| E(G)| \unicode{x02215}\lfloor | V(G)| \unicode{x02215}2\rfloor \gt {\rm{\Delta }}(G)$, and critical if χ′(H)<χ′(G)$\chi ^{\prime} (H)\lt \chi ^{\prime} (G)$ for every proper subgraph H$H$ of G$G$
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Partitioning kite-free planar graphs into two forests J. Graph Theory (IF 0.9) Pub Date : 2023-12-12 Yang Wang, Yiqiao Wang, Ko-Wei Lih
A kite is a complete graph on four vertices with one edge removed. It is proved that every planar graph without a kite as subgraph can be partitioned into two induced forests. This resolves a conjecture of Raspaud and Wang in 2008.
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Rainbow subgraphs in edge-colored complete graphs: Answering two questions by Erdős and Tuza J. Graph Theory (IF 0.9) Pub Date : 2023-12-12 Maria Axenovich, Felix C. Clemen
An edge-coloring of a complete graph with a set of colors C$C$ is called completely balanced if any vertex is incident to the same number of edges of each color from C$C$. Erdős and Tuza asked in 1993 whether for any graph F$F$ on ℓ$\ell $ edges and any completely balanced coloring of any sufficiently large complete graph using ℓ$\ell $ colors contains a rainbow copy of F$F$. This question was restated
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On Turán problems with bounded matching number J. Graph Theory (IF 0.9) Pub Date : 2023-12-06 Dániel Gerbner
Very recently, Alon and Frankl initiated the study of the maximum number of edges in n$n$-vertex F$F$-free graphs with matching number at most s$s$. For fixed F$F$ and s$s$, we determine this number apart from a constant additive term. We also obtain several exact results.
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Strengthening the directed Brooks' theorem for oriented graphs and consequences on digraph redicolouring J. Graph Theory (IF 0.9) Pub Date : 2023-12-06 Lucas Picasarri-Arrieta
Let D=(V,A)$D=(V,A)$ be a digraph. We define Δmax(D)${{\rm{\Delta }}}_{\max }(D)$ as the maximum of {max(d+(v),d−(v))∣v∈V}$\{\max ({d}^{+}(v),{d}^{-}(v))| v\in V\}$ and Δmin(D)${{\rm{\Delta }}}_{\min }(D)$ as the maximum of {min(d+(v),d−(v))∣v∈V}$\{\min ({d}^{+}(v),{d}^{-}(v))| v\in V\}$. It is known that the dichromatic number of D$D$ is at most Δmin(D)+1${{\rm{\Delta }}}_{\min }(D)+1$. In
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5-Coloring reconfiguration of planar graphs with no short odd cycles J. Graph Theory (IF 0.9) Pub Date : 2023-12-06 Daniel W. Cranston, Reem Mahmoud
The coloring reconfiguration graph Ck(G)${{\mathscr{C}}}_{k}(G)$ has as its vertex set all the proper k$k$-colorings of G$G$, and two vertices in Ck(G)${{\mathscr{C}}}_{k}(G)$ are adjacent if their corresponding k$k$-colorings differ on a single vertex. Cereceda conjectured that if an n$n$-vertex graph G$G$ is d$d$-degenerate and k≥d+2$k\ge d+2$, then the diameter of Ck(G)${{\mathscr{C}}}_{k}(G)$
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Heroes in oriented complete multipartite graphs J. Graph Theory (IF 0.9) Pub Date : 2023-12-01 Pierre Aboulker, Guillaume Aubian, Pierre Charbit
The dichromatic number of a digraph is the minimum size of a partition of its vertices into acyclic induced subgraphs. Given a class of digraphs C${\mathscr{C}}$, a digraph H$H$ is a hero in C${\mathscr{C}}$ if H$H$-free digraphs of C${\mathscr{C}}$ have bounded dichromatic number. In a seminal paper, Berger et al. give a simple characterisation of all heroes in tournaments. In this paper, we give
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Revisiting semistrong edge-coloring of graphs J. Graph Theory (IF 0.9) Pub Date : 2023-11-22 Borut Lužar, Martina Mockovčiaková, Roman Soták
A matching M $M$ in a graph G $G$ is semistrong if every edge of M $M$ has an endvertex of degree one in the subgraph induced by the vertices of M $M$ . A semistrong edge-coloring of a graph G $G$ is a proper edge-coloring in which every color class induces a semistrong matching. In this paper, we continue investigation of properties of semistrong edge-colorings initiated by Gyárfás and Hubenko. We
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A graph polynomial from chromatic symmetric functions J. Graph Theory (IF 0.9) Pub Date : 2023-11-23 William Chan, Logan Crew
Many graph polynomials may be derived from the coefficients of the chromatic symmetric function XG${X}_{G}$ of a graph G$G$ when expressed in different bases. For instance, the chromatic polynomial is obtained by mapping pn→x${p}_{n}\to x$ for each n$n$ in this function, while a polynomial whose coefficients enumerate acyclic orientations is obtained by mapping en→x${e}_{n}\to x$ for each n$n$. In
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K2-Hamiltonian graphs: II J. Graph Theory (IF 0.9) Pub Date : 2023-11-22 Jan Goedgebeur, Jarne Renders, Gábor Wiener, Carol T. Zamfirescu
In this paper, we use theoretical and computational tools to continue our investigation of K2${K}_{2}$-hamiltonian graphs, that is, graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, and their interplay with K1${K}_{1}$-hamiltonian graphs, that is, graphs in which every vertex-deleted subgraph is hamiltonian. Perhaps surprisingly, there exist graphs that are both
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Simple versus nonsimple loops on random regular graphs J. Graph Theory (IF 0.9) Pub Date : 2023-11-21 Benjamin Dozier, Jenya Sapir
In this note, we solve the “birthday problem” for loops on random regular graphs. Namely, for fixed d≥3$d\ge 3$, we prove that on a random d$d$-regular graph with n$n$ vertices, as n$n$ approaches infinity, with high probability: (i) almost all primitive nonbacktracking loops of length k≺n$k\prec \sqrt{n}$ are simple, that is, do not self-intersect, and (ii) almost all primitive nonbacktracking loops
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Induced subgraphs and tree decompositions V. one neighbor in a hole J. Graph Theory (IF 0.9) Pub Date : 2023-11-21 Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl, Kristina Vušković
What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the “basic treewidth obstructions”). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs
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Monochromatic spanning trees and matchings in ordered complete graphs J. Graph Theory (IF 0.9) Pub Date : 2023-11-15 János Barát, András Gyárfás, Géza Tóth
We study two well-known Ramsey-type problems for (vertex-)ordered complete graphs. Two independent edges in ordered graphs can be nested, crossing, or separated. Apart from two trivial cases, these relations define six types of subgraphs, depending on which one (or two) of these relations are forbidden. Our first target is to refine a remark by Erdős and Rado that every 2-coloring of the edges of a
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On the saturation spectrum of odd cycles J. Graph Theory (IF 0.9) Pub Date : 2023-11-09 Ronald J. Gould, André Kündgen, Minjung Kang
Given a graph H$H$, we say that a graph G$G$ is H$H$-saturated if H$H$ is not a subgraph of G$G$, but the addition of any new edge to G$G$ creates at least one copy of H$H$. In this paper we determine all pairs (n,m)$(n,m)$ for which there is a C5${C}_{5}$-saturated graph on n$n$ vertices and m$m$ edges. In addition, we determine all but O(nk)$O(nk)$ possible sizes for n$n$-vertex H$H$-saturated
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Inducibility in the hypercube J. Graph Theory (IF 0.9) Pub Date : 2023-11-09 John Goldwasser, Ryan Hansen
Let Qd${Q}_{d}$ be the hypercube of dimension d$d$ and let H$H$ and K$K$ be subsets of the vertex set V(Qd)$V({Q}_{d})$, called configurations in Qd${Q}_{d}$. We say that K$K$ is an exact copy of H$H$ if there is an automorphism of Qd${Q}_{d}$ which sends H$H$ onto K$K$. Let n≥d$n\ge d$ be an integer, let H$H$ be a configuration in Qd${Q}_{d}$ and let S$S$ be a configuration in Qn${Q}_{n}$. We let
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Rainbow powers of a Hamilton cycle in Gn,p J. Graph Theory (IF 0.9) Pub Date : 2023-11-05 Tolson Bell, Alan Frieze
We show that the threshold for having a rainbow copy of a power of a Hamilton cycle in a randomly edge colored copy of Gn,p${G}_{n,p}$ is within a constant factor of the uncolored threshold. Our proof requires (1+ε)$(1+\varepsilon )$ times the minimum number of colors.
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Transversals in regular uniform hypergraphs J. Graph Theory (IF 0.9) Pub Date : 2023-11-05 Michael A. Henning, Anders Yeo
The transversal number τ(H)$\tau (H)$ of a hypergraph H$H$ is the minimum number of vertices that intersect every edge of H$H$. This notion of transversal is fundamental in hypergraph theory and has been studied a great deal in the literature. A hypergraph H$H$ is r$r$-regular if every vertex of H$H$ has degree r$r$, that is, every vertex of H$H$ belongs to exactly r$r$ edges. Further, H$H$ is k$k$-uniform
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Elusive properties of infinite graphs J. Graph Theory (IF 0.9) Pub Date : 2023-10-26 Tamás Csernák, Lajos Soukup
A graph property is said to be elusive (or evasive) if every algorithm testing this property by asking questions of the form “is there an edge between vertices x $x$ and y $y$ ?” requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa–Karp–Rosenberg conjecture is that every nontrivial monotone graph property is elusive for any finite vertex set. We show that the situation
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Ear-decompositions, minimally connected matroids and rigid graphs J. Graph Theory (IF 0.9) Pub Date : 2023-10-26 Tibor Jordán
We prove that if the two-dimensional rigidity matroid of a graph G$G$ on at least seven vertices is connected, and G$G$ is minimal with respect to this property, then G$G$ has at most 3n−9$3n-9$ edges. This bound, which is best possible, extends Dirac's bound on the size of minimally 2-connected graphs to dimension two. The bound also sharpens the general upper bound of Murty for the size of minimally
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On Seymour's and Sullivan's second neighbourhood conjectures J. Graph Theory (IF 0.9) Pub Date : 2023-10-20 Jiangdong Ai, Stefanie Gerke, Gregory Gutin, Shujing Wang, Anders Yeo, Yacong Zhou
For a vertex x $x$ of a digraph, d + ( x ) ${d}^{+}(x)$ ( d − ( x ) ${d}^{-}(x)$ , respectively) is the number of vertices at distance 1 from (to, respectively) x $x$ and d + + ( x ) ${d}^{++}(x)$ is the number of vertices at distance 2 from x $x$ . In 1995, Seymour conjectured that for any oriented graph D $D$ there exists a vertex x $x$ such that d + ( x ) ≤ d + + ( x ) ${d}^{+}(x)\le {d}^{++}(x)$
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Uniformly 3-connected graphs J. Graph Theory (IF 0.9) Pub Date : 2023-10-17 Liqiong Xu
Let k$k$ be a positive integer. A graph is said to be uniformly k$k$-connected if between any pair of vertices the maximum number of independent paths is exactly k$k$. Dawes showed that all minimally 3-connected graphs can be constructed from K4${K}_{4}$ such that every graph in each intermediate step is also minimally 3-connected. In this paper, we generalize Dawes' result to uniformly 3-connected
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The Alon–Tarsi number of planar graphs revisited J. Graph Theory (IF 0.9) Pub Date : 2023-10-13 Yangyan Gu, Xuding Zhu
This paper gives a simple proof of the result that every planar graph G $G$ has Alon–Tarsi number at most 5, and has a matching M $M$ such that G − M $G-M$ has Alon–Tarsi number at most 4.
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On the list color function threshold J. Graph Theory (IF 0.9) Pub Date : 2023-10-12 Hemanshu Kaul, Akash Kumar, Jeffrey A. Mudrock, Patrick Rewers, Paul Shin, Khue To
The chromatic polynomial of a graph G $G$ , denoted P ( G , m ) $P(G,m)$ , is equal to the number of proper m $m$ -colorings of G $G$ . The list color function of graph G $G$ , denoted P ℓ ( G , m ) ${P}_{\ell }(G,m)$ , is a list analogue of the chromatic polynomial that has been studied since the early 1990s, primarily through comparisons with the corresponding chromatic polynomial. It is known that
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Brooks' theorem with forbidden colors J. Graph Theory (IF 0.9) Pub Date : 2023-10-11 Carl Johan Casselgren
We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if G $G$ is a connected graph with maximum degree Δ ( G ) ≥ 4 ${\rm{\Delta }}(G)\ge 4$ that is not a complete graph and P ⊆ V ( G ) $P\subseteq V(G)$ is a set of vertices where either
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On the structure of consistent cycles in cubic symmetric graphs J. Graph Theory (IF 0.9) Pub Date : 2023-10-09 Klavdija Kutnar, Dragan Marušič, Štefko Miklavič, Primož Šparl
A cycle in a graph is consistent if the automorphism group of the graph admits a one-step rotation of this cycle. A thorough description of consistent cycles of arc-transitive subgroups in the full automorphism groups of finite cubic symmetric graphs is given.