2,377 research outputs found
Triangle-free Uniquely 3-Edge Colorable Cubic Graphs
This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976
Triangle-free Uniquely 3-Edge Colorable Cubic Graphs
This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. Ā The only such graph previously known was given by Tutte in 1976
On Hamilton cycles in graphs defined by intersecting set systems
In 1970 Lov\'asz conjectured that every connected vertex-transitive graph
admits a Hamilton cycle, apart from five exceptional graphs. This conjecture
has recently been settled for graphs defined by intersecting set systems, which
feature prominently throughout combinatorics. In this expository article, we
retrace these developments and give an overview of the many different
ingredients in the proofs
Hamiltonian Strongly Regular Graphs
We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected non-Hamiltonian strongly regular graph on fewer than 99 vertices.Distance-regular graphs;Hamilton cycles JEL-code
Degree and neighborhood conditions for hamiltonicity of claw-free graphs
For a graph H , let Ļ t ( H ) = min { Ī£ i = 1 t d H ( v i ) | { v 1 , v 2 , ā¦ , v t } is an independent set in H } and let U t ( H ) = min { | ā i = 1 t N H ( v i ) | | { v 1 , v 2 , āÆ , v t } is an independent set in H } . We show that for a given number Ļµ and given integers p ā„ t \u3e 0 , k ā { 2 , 3 } and N = N ( p , Ļµ ) , if H is a k -connected claw-free graph of order n \u3e N with Ī“ ( H ) ā„ 3 and its RyjĆ”cĢekās closure c l ( H ) = L ( G ) , and if d t ( H ) ā„ t ( n + Ļµ ) ā p where d t ( H ) ā { Ļ t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p ā 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large:
(i) If k = 2 , Ī“ ( H ) ā„ 3 , and for a given t ( 1 ā¤ t ā¤ 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s ā„ 1 ). When t = 4 and d t ( H ) = Ļ 4 ( H ) , this proves a conjecture in Frydrych (2001).
(ii) If k = 3 , Ī“ ( H ) ā„ 24 , and for a given t ( 1 ā¤ t ā¤ 10 ) d t ( H ) \u3e t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved Ļ t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p ā 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer
Generation and Properties of Snarks
For many of the unsolved problems concerning cycles and matchings in graphs
it is known that it is sufficient to prove them for \emph{snarks}, the class of
nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part
of this paper we present a new algorithm for generating all non-isomorphic
snarks of a given order. Our implementation of the new algorithm is 14 times
faster than previous programs for generating snarks, and 29 times faster for
generating weak snarks. Using this program we have generated all non-isomorphic
snarks on vertices. Previously lists up to vertices have been
published. In the second part of the paper we analyze the sets of generated
snarks with respect to a number of properties and conjectures. We find that
some of the strongest versions of the cycle double cover conjecture hold for
all snarks of these orders, as does Jaeger's Petersen colouring conjecture,
which in turn implies that Fulkerson's conjecture has no small counterexamples.
In contrast to these positive results we also find counterexamples to eight
previously published conjectures concerning cycle coverings and the general
cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated
and typos corrected. This version differs from the published one in that the
Arxiv-version has data about the automorphisms of snarks; Journal of
Combinatorial Theory. Series B. 201
Developments on Spectral Characterizations of Graphs
In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs
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