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### A Short Proof of a Theorem Concerning Degree Sums and Connectivity on Hamiltonian Graphs

Bing Wei
1999 Journal of combinatorial theory. Series B (Print)
proved that if G is a 2-connected graph with n vertices such that d(u)+d(v)+d(w) n+} holds for any triple of independent vertices u, v, and w, then G is hamiltonian, where } is the vertex connectivity  ...  In this note, we will give a short proof of the above result. Academic Press  ...  The method we used in the proof may be applied in finding or proving other results which concern degree sums and the connectivity of graphs. In order to prove Theorem 2, we need the following.  ...

### Depth first search in claw-free graphs

Gábor Wiener
2017 Optimization Letters
The minimum degree of G is denoted by δ(G) and the minimum sum of degrees of k independent vertices of G is denoted by δ k (G).  ...  A graph G is traceable if it contains a hamiltonian path. The minimum leaf number ml(G) is the minimum number of leaves (vertices of degree 1) of the spanning trees of G.  ...  Let G be a connected claw-free graph of diameter at most 2 and let v be a non-cut vertex of G. Then there exists a hamiltonian path of G starting at v. Proof.  ...

### Dirac-type generalizations concerning large cycles in graphs

Zh.G. Nikoghosyan
2009 Discrete Mathematics
Bondy conjectured a common generalization of various results in hamiltonian graph theory concerning Hamilton and dominating cycles by introducing a notion of PD λ -cycles (cycles that dominate all paths  ...  We present the reverse version of this result including a theorem of Voss and Zuluaga as a special case.  ...  Acknowledgments Many thanks to the anonymous referees for their helpful comments and suggestions, which have considerably improved the abstract and the introduction.  ...

### On the minimum leaf number of cubic graphs [article]

Jan Goedgebeur, Kenta Ozeki, Nico Van Cleemput, Gábor Wiener
2018 arXiv   pre-print
We present new results concerning the minimum leaf number of cubic graphs: we show that if $G$ is a connected cubic graph of order $n$, then $\mathrm{ml}(G) \leq \frac{n}6 + \frac13$, improving on the  ...  The \emph{minimum leaf number} $\hbox{ml} (G)$ of a connected graph $G$ is defined as the minimum number of leaves of the spanning trees of $G$.  ...  Acknowledgements Most of the computations were carried out using the Stevin Supercomputer Infrastructure at Ghent University.  ...

### Page 2553 of Mathematical Reviews Vol. , Issue 84g [page]

1984 Mathematical Reviews
Brooks’s theorem states that a k-chromatic graph having no vertex of degree >k must have a subgraph which is a complete k-graph. Two short proofs of Brooks’s theorem appear in the book by B.  ...  If g=1, so that G is a 4-valent 3-connected simple graph on the torus, then the minimum of the face-valency-sums of its edges is at most 9, and can only be 9 if each edge is common to the boundaries of  ...

### Page 2093 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews
The author gives a sufficient condition, of degree-sum type, for a graph to be Hamiltonian.  ...  One of the many consequences of this closure theorem is that a 7-connected claw-free graph is Hamiltonian, a major step in proving the Matthews-Sumner con- jecture that a 4-connected claw-free graph is  ...

### More Progress on Tough Graphs - The Y2K Report

Doug Bauer, Hajo Broersma, Edward Schmeichel
2002 Electronic Notes in Discrete Mathematics
We now know that not every 2-tough graph is hamiltonian. In fact for every > 0, there exists a (9/4− ) -tough nontraceable graph.  ...  We continue our quadrennial survey of results that relate the toughness of a graph to its cycle structure.  ...  The degree sum condition can of course be converted to a weaker minimum degree condition. Theorem 13 Let G be a 1-tough graph on n ≥ 11 vertices with δ ≥ n−4 2 . Then G is hamiltonian. Faudree et al  ...

### Page 5191 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews
Francois Vanderseypen (Wijgmaal) 99h:05081 05C45 05C40 Wei, Bing [Wei, Bing'] (PRC-ASBJ-S; Beijing) A short proof of a theorem concerning degree sums and connectivity on Hamiltonian graphs.  ...  In this note, we give a short proof of the above result.”  ...

### Spanning Trees: A Survey

Kenta Ozeki, Tomoki Yamashita
2010 Graphs and Combinatorics
We mainly deal with spanning trees having some particular properties concerning a hamiltonian properties, for example, spanning trees with bounded degree, with bounded number of leaves, or with bounded  ...  In this paper, we give a survey of spanning trees.  ...  Acknowledgements The authors would like to thank the referees for their helpful comments and corrections.  ...

### Toughness in Graphs – A Survey

Douglas Bauer, Hajo Broersma, Edward Schmeichel
2006 Graphs and Combinatorics
We begin our discussion with a well-known theorem of Dirac [77]. Theorem 1. Let G be a graph on n ≥ 3 vertices with δ ≥ n 2 . Then G is hamiltonian.  ...  Toughness and Circumference In this section we survey results concerning the relationship between the toughness of a graph and its circumference.  ...  After simplifying the proof of Theorem 97 they used the result to obtain lower bounds on τ . Theorem 98. Let G be a connected graph.  ...

### Regular non-hamiltonian polyhedral graphs

Nico Van Cleemput, Carol T. Zamfirescu
2018 Applied Mathematics and Computation
Invoking Steinitz' Theorem, in the following a polyhedron shall be a 3-connected planar graph.  ...  Let G be a polyhedron of connectivity 3 and X The cubic case Tait conjectured [35] in 1884 that every cubic polyhedron is hamiltonian.  ...  Zamfirescu is supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO).  ...

### Page 1822 of Mathematical Reviews Vol. 53, Issue 6 [page]

1977 Mathematical Reviews
Theorem 2: For every n and k with n,k21, there exists an n-connected graph G such that XG* is non-Hamiltonian.  ...  The author proves that the above condition is sufficient for k=5 and provides a new, short proof of the previously established theorem that the conditions are sufficient for k=3 except for the case n=6  ...

### On some intriguing problems in hamiltonian graph theory—a survey

H.J. Broersma
2002 Discrete Mathematics
We survey results and open problems in hamiltonian graph theory centered around three themes: regular graphs, t-tough graphs, and claw-free graphs.  ...  The proof of Theorem 2.6 (via Theorems 2.8 and 2.9) uses several ideas from [17] , where a relatively short proof of (an extension of) Theorem 2.1 occurs.  ...  degree sums.  ...

### Supereulerian graphs: A survey

Paul A. Catlin
1992 Journal of Graph Theory
There is a reduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the  ...  total interval number of a graph.  ...  By Corollary 5.1A, Corollary 3.2A, and (a) of Theorem 2.2, the line graph of any 4-edge-connected graph is hamiltonian.  ...

### Page 2345 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews
The main result gives a minimum degree condition and a sum of degrees condition on a bipartite graph either of which implies the existence of a 2-factor with each cycle of the 2-factor containing a specified  ...  each of its longest cycles (without all being Hamiltonian).” 2002d:05077 05C38 05C35 Schrijver, Alexander (NL-MATH; Amsterdam) A short proof of Mader’s .7-paths theorem.  ...
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