Graphs Of Girth Research Articles

On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers

AbstractA graph is cubical if it is a subgraph of a hypercube. For a cubical graph H and a hypercube $$Q_n$$ Q n , $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) is the largest number of edges in an H-free subgraph of $$Q_n$$ Q n . If $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) is at least a positive proportion of the number of edges in $$Q_n$$ Q n , then H is said to have positive Turán density in the hypercube; otherwise it has zero Turán density. Determining $$\textrm{ex}(Q_n, H)$$ ex ( Q n , H ) and even identifying whether H has positive or zero Turán density remains a widely open question for general H. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge layer of some hypercube. Graphs H that are not layered have positive Turán density because one can form an H-free subgraph of $$Q_n$$ Q n consisting of edges of every other layer. For example, a 4-cycle is not layered and has positive Turán density. However, in general, it is not obvious what properties layered graphs have. We give a characterization of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least 12 and of length 8. However, we prove that there are cubical graphs of girth 8 that are not layered and thus having positive Turán density. The cycle of length 10 remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that $$\textrm{ex}(Q_n, C_{10})= \Omega (n2^n/ \log ^a n)$$ ex ( Q n , C 10 ) = Ω ( n 2 n / log a n ) , for a constant a, showing that the extremal number for a 10-cycle behaves differently from any other cycle of zero Turán density.

Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles

AbstractIn this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let denote the maximum number of edge‐disjoint copies of a fixed simple graph that can be placed on an ‐vertex ground set without forming a subgraph whose edges are from different ‐copies. The case when both and are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when and are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl, and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear ‐uniform hypergraph Turán problems to determine form a class of the multicolor Turán problem, following the identity , our results determine the linear hypergraph Turán numbers of every graph of girth 3 and for every up to a subpolynomial factor. Furthermore, when is a triangle, we settle the case and give bounds for the cases , as well.

Partitioning planar graph of girth 5 into two forests with maximum degree 4

Partitioning planar graph of girth 5 into two forests with maximum degree 4

Density of 3‐critical signed graphs

AbstractWe say that a signed graph is ‐critical if it is not ‐colorable but every one of its proper subgraphs is ‐colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3‐critical signed graph on vertices has at least edges, and that this bound is asymptotically tight. It follows that every signed planar or projective‐planar graph of girth at least 6 is (circular) 3‐colorable, and for the projective‐planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph , which is the positive triangle augmented with a negative loop on each vertex.

Decompositions into two linear forests of bounded lengths

For some k∈Z≥0∪{∞}, we call a linear forest k-bounded if each of its components has at most k edges. We will say a (k,ℓ)-bounded linear forest decomposition of a graph G is a partition of E(G) into the edge sets of two linear forests Fk,Fℓ where Fk is k-bounded and Fℓ is ℓ-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both k and ℓ are at least 2, NP-complete if k≥9 and ℓ=1, and is in P for (k,ℓ)=(2,1). Before this, the only known NP-complete cases were the (2,2) and (3,3) cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than 3-edge-coloring such graphs.

The square of every subcubic planar graph of girth at least 6 is 7-choosable

The square of a graph G, denoted G2, has the same vertex set as G and has an edge between two vertices if the distance between them in G is at most 2. Thomassen [16] and Hartke, Jahanbekam and Thomas [8] proved that χ(G2)≤7 if G is a subcubic planar graph. A natural question is whether χℓ(G2)≤7 or not if G is a subcubic planar graph. It was showed in [5] that χℓ(G2)≤7 if G is a subcubic planar graph of girth at least 7. We prove that χℓ(G2)≤7 if G is a subcubic planar graph of girth at least 6.

Weak Degeneracy of Planar Graphs and Locally Planar Graphs

Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d+1)$-list assignment $L$ of $G$, one can construct an $L$ coloring of $G$ by a modified greedy coloring algorithm. It is known that planar graphs of girth 5 are 3-choosable and locally planar graphs are $5$-choosable. This paper strengthens these results and proves that planar graphs of girth 5 are weakly 2-degenerate and locally planar graphs are weakly 4-degenerate.

On decreasing the orders of (k,g)-graphs

A (k,g)-graph is a k-regular graph of girth g. Given kge 2 and gge 3, (k,g)-graphs of infinitely many orders are known to exist and the problem of finding a (k, g)-graph of the smallest possible order is known as the Cage Problem. The aim of our paper is to develop systematic (programmable) ways for lowering the orders of existing (k,g)-graphs, while preserving their regularity and girth. Such methods, in analogy with the previously used excision, may have the potential for constructing smaller (k, g)-graphs from current smallest examples—record holders—some of which have not been improved in years. In addition, we consider constructions that preserve the regularity, the girth and the order of the considered graphs, but alter the graphs enough to possibly make them suitable for the application of our order decreasing methods. We include a detailed discussion of several specific parameter cases for which several non-isomorphic smallest examples are known to exist, and address the question of the distance between these non-isomorphic examples based on the number of changes required to move from one example to another.

Token Sliding on Graphs of Girth Five

In the Token Sliding problem we are given a graph G and two independent sets Is\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_s$$\\end{document} and It\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_t$$\\end{document} in G of size k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ge 1$$\\end{document}. The goal is to decide whether there exists a sequence ⟨I1,I2,…,Iℓ⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle I_1, I_2, \\ldots , I_\\ell \\rangle $$\\end{document} of independent sets such that for all j∈{1,…,ℓ-1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j \\in \\{1,\\ldots , \\ell - 1\\}$$\\end{document} the set Ij\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_j$$\\end{document} is an independent set of size k, I1=Is\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_1 = I_s$$\\end{document}, Iℓ=It\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_\\ell = I_t$$\\end{document} and Ij▵Ij+1={u,v}∈E(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_j \ riangle I_{j + 1} = \\{u, v\\} \\in E(G)$$\\end{document}. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the problem asks whether there exists a sequence of independent sets that transforms Is\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_s$$\\end{document} into It\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I_t$$\\end{document} where at each step we are allowed to slide one token from a vertex to a neighboring vertex. In this paper, we focus on the parameterized complexity of Token Sliding parameterized by k. As shown by Bartier et al. (Algorithmica 83(9):2914–2951, 2021. https://doi.org/10.1007/s00453-021-00848-1), the problem is W[1]-hard on graphs of girth four or less, and the authors posed the question of whether there exists a constant p≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\ge 5$$\\end{document} such that the problem becomes fixed-parameter tractable on graphs of girth at least p. We answer their question positively and prove that the problem is indeed fixed-parameter tractable on graphs of girth five or more, which establishes a full classification of the tractability of Token Sliding parameterized by the number of tokens based on the girth of the input graph.

Computational complexity aspects of super domination

Let G be a graph. A dominating set D⊆V(G) is a super dominating set if for every vertex x∈V(G)∖D there exists y∈D such that NG(y)∩(V(G)∖D))={x}. The cardinality of a smallest super dominating set of G is the super domination number of G. An exact formula for the super domination number of a tree T is obtained, and it is demonstrated that a smallest super dominating set of T can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph G is at most a given integer if G is a bipartite graph of girth at least 8. The super domination number is determined for all k-subdivisions of graphs. Interestingly, in half of the cases the exact value can be efficiently computed from the obtained formulas, while in the other cases the computation is hard. While obtaining these formulas, II-matching numbers are introduced and proved that they are computationally hard to determine.

Bounds and Extremal Graphs for Total Dominating Identifying Codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$.
 Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached).
 Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order $n$ has a total dominating identifying code of size at most $\frac{3n}{4}$. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this upper bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains the upper bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins).
 Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the similar parameter $\gamma^{\text{ID}}(G)$ as well as to the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.

Finding geometric representations of apex graphs is NP-hard

Planar graphs can be represented as the intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., SODA 2018). For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex graphs, i.e., graphs that can be made planar by removing one vertex from them. We show, somewhat surprisingly, that deciding whether geometric representations exist for apex graphs is NP-hard as well.More precisely, we show that for every fixed positive integer g and every graph class G such that ▪, it is NP-hard to decide whether an input graph belongs to the graph class G, even when the inputs are restricted to apex graphs of girth g. Here, ▪ is the class of intersection graphs of axis-parallel line segments (where horizontal segments intersect only vertical segments), and ▪ is the class of intersection graphs of simple curves (where two intersecting curves cross each other exactly once) in the plane. This partially answers an open question raised by Kratochvíl & Pergel (COCOON, 2007).Most known reductions for earlier proofs of NP-hardness for these problems are from variants of 3-SAT (mainly Planar 3-Connected 3-SAT). We reduce from the ▪▪▪ problem, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric intersection graphs.

On tetravalent half-arc-transitive graphs of girth 5

A subgroup of the automorphism group of a graph Γ is said to be half-arc-transitive on Γ if its action on Γ is transitive on the vertex set of Γ and on the edge set of Γ but not on the arc set of Γ. Tetravalent graphs of girths 3 and 4 admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth 5. We show that, with two exceptions, all such graphs only have directed 5-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed 5-cycles, study some of their graph theoretic properties and prove that the 5-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth 5 admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth 5.

Locally irregular edge-coloring of subcubic graphs

A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular subgraph. Among the graphs admitting a locally irregular edge-coloring, i.e., decomposable graphs, only one is known to require 4 colors, while for all the others it is believed that 3 colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree 3, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most 3 colors. We also discuss when 2 colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth 4 which require 3 colors.

Cubic Vertex-Transitive Graphs Admitting Automorphisms of Large Order

A connected graph of order n admitting a semiregular automorphism of order n/k is called a k-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper, we study the broader class of cubic vertex-transitive graphs of order n admitting an automorphism of order n/3 or larger that may not be semiregular. In particular, we show that any such graph is either a k-multicirculant for some k le 3, or it belongs to an infinite family of graphs of girth 6.

Resolution of a conjecture about linking ring structures

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Potočnik and S. Wilson (2014) [12], as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper.

Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree

AbstractGiven a graph of degree over vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth , we develop a local message passing algorithm whose complexity is , and that achieves near optimal cut values among all ‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value , where is the value of the Parisi formula from spin glass theory, and (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes after removing a small fraction of vertices. Earlier work established that, for random ‐regular graphs, the typical max‐cut value is . Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs.

Small Graphs and Hypergraphs of Given Degree and Girth

The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a $d$-regular, $r$-uniform hypergraph of given (Berge) girth $g$. We show that these two problems are in fact very closely linked. By extending the ideas of Cayley graphs to the hypergraph context, we find smallest known hypergraphs for various parameter sets. Because of the close link to the cage problem from graph theory, we are able to use these techniques to find new record smallest cubic graphs of girths 23, 24, 28, 29, 30, 31 and 32.

Recoloring Planar Graphs of Girth at Least Five

For a positive integer , the -recoloring graph of a graph has as vertex set all proper -colorings of with two -colorings being adjacent if they differ by the color of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the -recoloring graphs of planar graphs, the 5-recoloring graphs of triangle-free planar graphs and the 4-recoloring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose 6-recoloring graph is disconnected, triangle-free planar graphs whose 4-recoloring graph is disconnected, and planar graphs of any given girth whose 3-recoloring graph is disconnected. The main result of this paper consists in showing, via a novel application of the discharging method, that the 4-recoloring graph of every planar graph of girth five is connected. This completes the classification of the connectedness of the recoloring graph for planar graphs of given girth. We also prove some theorems regarding the diameter of the recoloring graph of planar graphs.

Contractible edges in longest cycles

AbstractThis paper investigates the number of contractible edges in a longest cycle of a ‐connected graph that is triangle‐free or has minimum degree at least . We prove that, except for two graphs, contains at least contractible edges. For triangle‐free 3‐connected graphs, we show that contains at least contractible edges, and characterize all graphs having a longest cycle containing exactly six/seven contractible edges. Both results are tight. Lastly, we prove that every longest cycle of a 3‐connected graph of girth at least 5 contains at least contractible edges.