Global properties of graphs with local degree conditions

2016

A nonhamiltonian LLH graph of order 15 with maximum degree 14.. 4.5 The graph N (u 1) ∩ N (u 2) ∩ • • • ∩ N (u k) used in the proof of Lemma 4.

Časopis pro pěstování matematiky, 1974

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz časopІs pro pëstovánf matematiky, roč. 99 (1974), Praha LOCALLY CONNECTED GRAPHS

Israel Journal of Mathematics, 1979

A graph G is locally n-connected, n >_-1, if the subgraph induced by the neighborhood of each vertex is n-connected. It is shown that every connected, locally 3-connected graph containing no induced subgraph isomorphic to K(1, 3) is h amilt onian-connected.

Discrete Mathematics, 2009

For any positive integer k and any (2 + k − n)-connected graph of order n, we define, following Bondy and Chvatàl, the k-neighborhood closure NC k (G) as the graph obtained from G by recursively joining pairs of nonadjacent vertices a, b satisfying the condition |N(a) ∪ N(b)| + δ ab + ε ab ≥ k, where δ ab = min {d(x)|a, b ∈ N(x) ∪ {x}} and ε ab is a well defined binary variable. For many properties P of G, there exists a suitable k (depending on P and n) such that NC k (G) has property P if and only if G does.

We consider the existence of hamiltonian cycles for locally connected graphs with a bounded vertex degree. For a graph G, let ∆(G) and δ(G) denote the maximum and minimum vertex degrees, respectively. We explicitly describe all connected, locally connected graphs with ∆(G) 4. We show that every connected, locally connected graph with ∆(G) = 5 and δ(G) 3 is fully cycle extendable which extends the results of P.B. Kikust (Latvian Math. Annual 16 (1975) 33-38) and G.R.T. Hendry (J. Graph Theory 13 (1989) 257-260) on fully cycle extendability of connected, locally connected graphs with the maximum vertex degree bounded by 5. Furthermore, we prove that problem Hamilton Cycle for locally connected graphs with ∆(G) 7 is NP-complete.

Journal of Graph Theory, 2006

Dirac proved that a graph G is hamiltonian if the minimum degree δ(G) ≥ n/2, where n is the order of G. Let G be a graph and A ⊆ V(G). The neighborhood of A is N(A) = {b : ab ∈ E(G) for some a ∈ A}. For any positive integer k, we show that every (2k − 1)-connected graph of order n ≥ 16k 3 is hamiltonian if |N(A)| ≥ n/2 for every independent vertex set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k − 1 on connectivity is best possible in general while the lower bound 16k 3 for n is conjectured to be unnecessary.

Discrete Mathematics, 1992

Dirac proved that if each vertex of a graph G of order n⩾3 has degree at least n/2, then the graph is Hamiltonian. This result will be generalized by showing that if the union of the neighborhoods of each pair of vertices of a 2-connected graph G of sufficiently large order n has at least n/2 vertices, then G is Hamiltonian. Other results that are based on neighborhood unions of pairs of vertices will be proved that give the existence of cycles, paths and matchings. Also, Hamiltonian results will be considered that use the union of neighborhoods of more than 2 vertices.

Discrete Applied Mathematics, 2016

Research interests: Structural graph theory with emphasis on (i) distance notions in graphs including graph convexity and the metric dimension in graphs, (ii) graph connectivity, (iii) local structure versus global structure, including Ryjacek's conjecture and (iv) the path partition conjecture.

The Electronic Journal of Combinatorics, 2018

For a graph $G$, let $\chi(G)$ and $\lambda(G)$ denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac implies that every graph $G$ satisfies $\chi(G)\leq \lambda(G)+1$. In this paper we characterize the graphs $G$ for which $\chi(G)=\lambda(G)+1$. The case $\lambda(G)=3$ was already solved by Aboulker, Brettell, Havet, Marx, and Trotignon. We show that a graph $G$ with $\lambda(G)=k\geq 4$ satisfies $\chi(G)=k+1$ if and only if $G$ contains a block which can be obtained from copies of $K_{k+1}$ by repeated applications of the Hajós join.

Discrete Mathematics, 1999