21,033 research outputs found
Graphs that do not contain a cycle with a node that has at least two neighbors on it
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 -index of minimally -(edge-)connected graphs for small
Let be a graph with adjacency matrix and let be the
diagonal matrix of vertex degrees of . For any real ,
Nikiforov defined the -matrix of a graph as . The largest eigenvalue of is called the
-index or the -spectral radius of . A graph is minimally
-(edge)-connected if it is -(edge)-connected and deleting any arbitrary
chosen edge always leaves a graph which is not -(edge)-connected. In this
paper, we characterize the minimally 2-edge-connected graphs and minimally
3-connected graph with given order having the maximum -index for
, respectively.Comment: arXiv admin note: text overlap with arXiv:2301.0338
Connectivity and spanning trees of graphs
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
A {\em strong edge coloring} of a graph 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 is the minimum number of colors in a
strong edge coloring of . 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 -degenerate graph is at most
. As a direct consequence, the strong
chromatic index of a -degenerate graph is at most ,
which improves the upper bound by Chang and Narayanan
[Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2)
119--126]. For a special subclass of -degenerate graphs, we obtain a better
upper bound, namely if is a graph such that all of its -vertices
induce a forest, then \chiup_{s}'(G) \leq 4 \Delta(G) -3; as a corollary,
every minimally -connected graph has strong chromatic index at most . 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
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 denoted
by , is the minimum positive integer such that has an acyclic
edge coloring with colors. It has been conjectured by Fiam\v{c}\'{\i}k that
for any graph with maximum degree . Linear
arboricity of a graph , denoted by , is the minimum number of linear
forests into which the edges of can be partitioned. A graph is said to be
chordless if no cycle in the graph contains a chord. Every -connected
chordless graph is a minimally -connected graph. It was shown by Basavaraju
and Chandran that if is -degenerate, then . Since
chordless graphs are also -degenerate, we have for any
chordless graph . Machado, de Figueiredo and Trotignon proved that the
chromatic index of a chordless graph is when . 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 , except when and the graph has a cycle, in which
case it is . 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 , unless
has a cycle with , in which case . 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
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
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