21,033 research outputs found

    Graphs that do not contain a cycle with a node that has at least two neighbors on it

    Get PDF
    We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring

    On the α\alpha-index of minimally kk-(edge-)connected graphs for small kk

    Full text link
    Let GG be a graph with adjacency matrix A(G)A(G) and let D(G)D(G) be the diagonal matrix of vertex degrees of GG. For any real α∈[0,1]\alpha \in [0,1], Nikiforov defined the AαA_\alpha-matrix of a graph GG as Aα(G)=αD(G)+(1−α)A(G)A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G). The largest eigenvalue of Aα(G)A_\alpha(G) is called the α\alpha-index or the AαA_\alpha-spectral radius of GG. A graph is minimally kk-(edge)-connected if it is kk-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not kk-(edge)-connected. In this paper, we characterize the minimally 2-edge-connected graphs and minimally 3-connected graph with given order having the maximum α\alpha-index for α∈[12,1)\alpha \in [\frac{1}{2},1), respectively.Comment: arXiv admin note: text overlap with arXiv:2301.0338

    Connectivity and spanning trees of graphs

    Get PDF
    This dissertation focuses on connectivity, edge connectivity and edge-disjoint spanning trees in graphs and hypergraphs from the following aspects.;1. Eigenvalue aspect. Let lambda2(G) and tau( G) denote the second largest eigenvalue and the maximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of tau(G), Cioaba and Wong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if lambda 2(G)) \u3c d -- 2k-1d+1 , then tau(G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. We propose a more general conjecture that for a graph G with minimum degree delta ≥ 2 k ≥ 4, if lambda2(G) \u3c delta -- 2k-1d+1 then tau(G) ≥ k. We prove the conjecture for k = 2, 3 and provide partial results for k ≥ 4. We also prove that for a graph G with minimum degree delta ≥ k ≥ 2, if lambda2( G) \u3c delta -- 2k-1d +1 , then the edge connectivity is at least k. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on tau(G) and edge connectivity.;2. Network reliability aspect. With graphs considered as natural models for many network design problems, edge connectivity kappa\u27(G) and maximum number of edge-disjoint spanning trees tau(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G. Let kappa\u27(G) = max{lcub}kappa\u27(H) : H is a subgraph of G{rcub}. We present: (i) For each integer k \u3e 0, a characterization for graphs G with the property that kappa\u27(G) ≤ k but for any additional edge e not in G, kappa\u27(G + e) ≥ k + 1. (ii) For any integer n \u3e 0, a characterization for graphs G with |V(G)| = n such that kappa\u27(G) = tau( G) with |E(G)| minimized.;3. Generalized connectivity. For an integer l ≥ 2, the l-connectivity kappal( G) of a graph G is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Let k ≥ 1, a graph G is called (k, l)-connected if kappa l(G) ≥ k. A graph G is called minimally (k, l)-connected if kappal(G) ≥ k but ∀e ∈ E( G), kappal(G -- e) ≤ k -- 1. A structural characterization for minimally (2, l)-connected graphs and some extremal results are obtained. These extend former results by Dirac and Plummer on minimally 2-connected graphs.;4. Degree sequence aspect. An integral sequence d = (d1, d2, ···, dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r- uniform hypergraph with degree sequence d. It is proved that an r-uniform hypergraphic sequence d = (d1, d2, ···, dn) has a k-edge-connected realization if and only if both di ≥ k for i = 1, 2, ···, n and i=1ndi≥ rn-1r-1 , which generalizes the formal result of Edmonds for graphs and that of Boonyasombat for hypergraphs.;5. Partition connectivity augmentation and preservation. Let k be a positive integer. A hypergraph H is k-partition-connected if for every partition P of V(H), there are at least k(| P| -- 1) hyperedges intersecting at least two classes of P. We determine the minimum number of hyperedges in a hypergraph whose addition makes the resulting hypergraph k-partition-connected. We also characterize the hyperedges of a k-partition-connected hypergraph whose removal will preserve k-partition-connectedness

    Strong chromatic index of k-degenerate graphs

    Full text link
    A {\em strong edge coloring} of a graph GG is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} \chiup_{s}'(G) of a graph GG is the minimum number of colors in a strong edge coloring of GG. In this note, we improve a result by D{\k e}bski \etal [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show that the strong chromatic index of a kk-degenerate graph GG is at most (4k−2)⋅Δ(G)−2k2+1(4k-2) \cdot \Delta(G) - 2k^{2} + 1. As a direct consequence, the strong chromatic index of a 22-degenerate graph GG is at most 6Δ(G)−76\Delta(G) - 7, which improves the upper bound 10Δ(G)−1010\Delta(G) - 10 by Chang and Narayanan [Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2) 119--126]. For a special subclass of 22-degenerate graphs, we obtain a better upper bound, namely if GG is a graph such that all of its 3+3^{+}-vertices induce a forest, then \chiup_{s}'(G) \leq 4 \Delta(G) -3; as a corollary, every minimally 22-connected graph GG has strong chromatic index at most 4Δ(G)−34 \Delta(G) - 3. Moreover, all the results in this note are best possible in some sense.Comment: 3 pages in Discrete Mathematics, 201

    Acyclic Chromatic Index of Chordless Graphs

    Full text link
    An acyclic edge coloring of a graph is a proper edge coloring in which there are no bichromatic cycles. The acyclic chromatic index of a graph GG denoted by a′(G)a'(G), is the minimum positive integer kk such that GG has an acyclic edge coloring with kk colors. It has been conjectured by Fiam\v{c}\'{\i}k that a′(G)≤Δ+2a'(G) \le \Delta+2 for any graph GG with maximum degree Δ\Delta. Linear arboricity of a graph GG, denoted by la(G)la(G), is the minimum number of linear forests into which the edges of GG can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. Every 22-connected chordless graph is a minimally 22-connected graph. It was shown by Basavaraju and Chandran that if GG is 22-degenerate, then a′(G)≤Δ+1a'(G) \le \Delta+1. Since chordless graphs are also 22-degenerate, we have a′(G)≤Δ+1a'(G) \le \Delta+1 for any chordless graph GG. Machado, de Figueiredo and Trotignon proved that the chromatic index of a chordless graph is Δ\Delta when Δ≥3\Delta \ge 3. They also obtained a polynomial time algorithm to color a chordless graph optimally. We improve this result by proving that the acyclic chromatic index of a chordless graph is Δ\Delta, except when Δ=2\Delta=2 and the graph has a cycle, in which case it is Δ+1\Delta+1. We also provide the sketch of a polynomial time algorithm for an optimal acyclic edge coloring of a chordless graph. As a byproduct, we also prove that la(G)=⌈Δ2⌉la(G) = \lceil \frac{\Delta }{2} \rceil, unless GG has a cycle with Δ=2\Delta=2, in which case la(G)=⌈Δ+12⌉=2la(G) = \lceil \frac{\Delta+1}{2} \rceil = 2. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado, de Figueiredo and Trotignon for this class of graphs. This might be of independent interest

    Extremal Infinite Graph Theory

    Get PDF
    We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
    • …
    corecore