Degree Conditions for H-Linked Digraphs

Given a (multi)digraph H, a digraph D is H-linked if every injective function ι:V(H) → V(D) can be extended to an H-subdivision. In this paper, we give sharp degree conditions that ensure a sufficiently large digraph D is H-linked for arbitrary H. The notion of an H-linked digraph extends the classes of m-linked, m-ordered and strongly m-connected digraphs.First, we give sharp minimum semi-degree conditions for H-linkedness, extending results of Kühn and Osthus on m-linked and m-ordered digraphs. It is known that the minimum degree threshold for an undirected graph to be H-linked depends on a partition of the (undirected) graph H into three parts. Here, we show that the corresponding semi-degree threshold for H-linked digraphs depends on a partition of H into as many as nine parts.We also determine sharp Ore–Woodall-type degree-sum conditions ensuring that a digraph D is H-linked for general H. As a corollary, we obtain (previously undetermined) sharp degree-sum conditions for m-linked and m-ordered digraphs.

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Sharp Minimum Degree Conditions for the Existence of Disjoint Theta Graphs

The Electronic Journal of Combinatorics ◽

10.37236/9670 ◽

2021 ◽

Vol 28 (3) ◽

Author(s):

Emily Marshall ◽

Michael Santana

Keyword(s):

Minimum Degree ◽

Degree Condition ◽

Sharp Minimum ◽

Disjoint Cycles ◽

Vertex Graph ◽

Degree Conditions ◽

Delta G

In 1963, Corrádi and Hajnal showed that if $G$ is an $n$-vertex graph with $n \ge 3k$ and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles; furthermore, this result is best possible, both in terms of the number of vertices as well as the minimum degree. In this paper we focus on an analogue of this result for theta graphs. Results from Kawarabayashi and Chiba et al. showed that if $n = 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k \rceil$, or if $n$ is large with respect to $k$ and $\delta(G) \ge 2k+1$, respectively, then $G$ contains $k$ disjoint theta graphs. While the minimum degree condition in both results are sharp for the number of vertices considered, this leaves a gap in which no sufficient minimum degree condition is known. Our main result in this paper resolves this by showing if $n \ge 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k\rceil$, then $G$ contains $k$ disjoint theta graphs. Furthermore, we show this minimum degree condition is sharp for more than just $n = 4k$, and we discuss how and when the sharp minimum degree condition may transition from $\lceil \frac{5}{2}k\rceil$ to $2k+1$.

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Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

The Electronic Journal of Combinatorics ◽

10.37236/3610 ◽

2015 ◽

Vol 22 (4) ◽

Cited By ~ 1

Author(s):

Louis DeBiasio ◽

Theodore Molla

Keyword(s):

Directed Graph ◽

Hamiltonian Cycle ◽

Minimum Degree ◽

Directed Graphs ◽

Hamiltonian Cycles ◽

Alternate Direction ◽

Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

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Bounds for the Generalized Distance Eigenvalues of a Graph

Symmetry ◽

10.3390/sym11121529 ◽

2019 ◽

Vol 11 (12) ◽

pp. 1529 ◽

Cited By ~ 5

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal Ahmad Ganie ◽

Yilun Shang

Keyword(s):

Spectral Radius ◽

Undirected Graph ◽

Minimum Degree ◽

Distance Matrix ◽

Laplacian Matrix ◽

Extremal Graphs ◽

Special Cases ◽

Signless Laplacian ◽

Graph Parameters ◽

Generalized Distance

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

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Pitchfork domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500251 ◽

2020 ◽

Vol 12 (02) ◽

pp. 2050025

Author(s):

Manal N. Al-Harere ◽

Mohammed A. Abdlhusein

Keyword(s):

Maximum Degree ◽

Undirected Graph ◽

Minimum Degree ◽

Dominating Set ◽

Domination Number ◽

Dominating Sets ◽

Minimum Cardinality ◽

New Model ◽

Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

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Maximum and Minimum Degree Conditions for Embedding Trees

SIAM Journal on Discrete Mathematics ◽

10.1137/19m1277667 ◽

2020 ◽

Vol 34 (4) ◽

pp. 2108-2123

Author(s):

Guido Besomi ◽

Matías Pavez-Signé ◽

Maya Stein

Keyword(s):

Minimum Degree ◽

Degree Conditions ◽

Maximum And Minimum Degree

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NOTE ON MINIMUM DEGREE AND PROPER CONNECTION NUMBER

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720000593 ◽

2020 ◽

pp. 1-5

Author(s):

YUEYU WU ◽

YUNQING ZHANG ◽

YAOJUN CHEN

Keyword(s):

Minimum Degree ◽

Connection Number ◽

Minimum Value ◽

Proper Connection Number ◽

Degree Conditions

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number, $pc(G)$ , of a graph $G$ , is the smallest number of colours that are needed to colour $G$ such that it is properly connected. Let $\unicode[STIX]{x1D6FF}(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\unicode[STIX]{x1D6FF}(n)$ . Brause et al. [‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin.33 (2017), 833–843] showed that $\unicode[STIX]{x1D6FF}(n)>n/42$ . In this note, we show that $\unicode[STIX]{x1D6FF}(n)>n/36$ .

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Rainbow connection numbers and the minimum degree sum of a graph

Scientia Sinica Mathematica ◽

10.1360/012012-120 ◽

2013 ◽

Vol 43 (1) ◽

pp. 7-14 ◽

Cited By ~ 1

Author(s):

XueLiang LI ◽

JiuYing DONG

Keyword(s):

Minimum Degree ◽

Degree Sum ◽

Rainbow Connection

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A Parameterized Complexity View on Collapsing k-Cores

Theory of Computing Systems ◽

10.1007/s00224-021-10045-w ◽

2021 ◽

Author(s):

Junjie Luo ◽

Hendrik Molter ◽

Ondřej Suchý

Keyword(s):

Parameterized Complexity ◽

Undirected Graph ◽

Minimum Degree ◽

Structural Parameters ◽

Vertex Cover ◽

Polynomial Kernel ◽

Input Graph ◽

Induced Subgraph ◽

Vertex Deletion ◽

Drop Outs

AbstractWe study the -hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. (2017) and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be -hard for all r ≥ 0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r. We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k ≤ 2 and k ≥ 3. For the latter case it is known that for all x ≥ 0 Collapsed k-Core is -hard when parameterized by b. For k ≤ 2 we show that Collapsed k-Core is -hard when parameterized by b and in when parameterized by (b + x). Furthermore, we outline that Collapsed k-Core is in when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

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