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Triangle-free eulerian tours in graphs with maximum degree at most 4

Tobias Adelgren
1995 Discrete Mathematics  
In this paper we completely characterise the family of eulerian simple graphs G with maximum degree at most 4 which admit a triangle-free eulerian tour, i.e., a sequence v l v2 .  ...  --v" c l such that each vi is a vertex, the pairs vi, vi+ 1, i = 1,2 ..... m are the m distinct edges in G and finally, v~+ 3 ~ v~ for all i = 1,2 ..... m, with indices counted modulo m. 0012-365X/95/$09.50  ...  The graphs in J~ are the only ones to obstruct a triangle-free euler tour Jbr graphs with maximum degree at most 4.  ... 
doi:10.1016/0012-365x(94)00183-j fatcat:yr3pcetvhzfufe5y4v7nfpu4vy

Page 7185 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
Eulerian tours in graphs with maximum degree at most 4.  ...  Summary: “In this paper we completely characterise the family of Eulerian simple graphs G with maximum degree at most 4 which admit a triangle-free Eulerian tour, i.e., a sequence v2 --- Um} such that  ... 

Triangle-Free Circuit Decompositions and Petersen Minor

Katherine Heinrich, Jiping Liu, Cun-Quan Zhang
1998 Journal of combinatorial theory. Series B (Print)  
In this paper, we characterize those simple graphs with no Petersen minor which admit triangle-free circuit decompositions. 1998 Academic Press  ...  In [6] Heinrich, Liu, and Yu characterized graphs of maximum degree 4 which admit triangle-free eulerian tours (that is, no three consecutive edges on the tour form a triangle); but these tours may not  ...  If we attach a triangle to each edge in the perfect matching, we obtain infinitely many eulerian graphs with degrees 2 and 4, which are not 2-graphs and do not have trianglefree circuit decomposition,  ... 
doi:10.1006/jctb.1997.1808 fatcat:rdc7mawitvhofkxc6eide63wnm

An improved upper bound for the TSP in cubic 3-edge-connected graphs

David Gamarnik, Moshe Lewenstein, Maxim Sviridenko
2005 Operations Research Letters  
Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP in general.  ...  We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs.  ...  Another possibility is to obtain improved results for sparse graphs, say graphs with maximum degree 3. At this point, we cannot even lift the 3-connectivity restriction.  ... 
doi:10.1016/j.orl.2004.09.005 fatcat:fnzbstj7qbby5l6fjzyuzixeqy

TSP on Cubic and Subcubic Graphs [chapter]

Sylvia Boyd, René Sitters, Suzanne van der Ster, Leen Stougie
2011 Lecture Notes in Computer Science  
Using polyhedral techniques in an interesting way, we obtain a polynomial-time 4/3-approximation algorithm for this problem on cubic graphs, improving upon Christofides' 3/2-approximation, and upon the  ...  We also prove that, as an upper bound, the 4/3 conjecture is true for this problem on cubic graphs.  ...  Every 2-edge connected graph of maximum degree 3 has a TSP tour of length at most 7 5 n − 4 5 . As with the cubic case, this result can be extended to include bridges.  ... 
doi:10.1007/978-3-642-20807-2_6 fatcat:donyaxftkfcmpd44bbthuyueay

The traveling salesman problem on cubic and subcubic graphs

Sylvia Boyd, René Sitters, Suzanne van der Ster, Leen Stougie
2012 Mathematical programming  
In fact we prove constructively that for any cubic graph on n vertices a tour of length 4n/3 − 2 exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP.  ...  We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest.  ...  maximum degree is 6.  ... 
doi:10.1007/s10107-012-0620-1 fatcat:34lq2cwdbrfqzntudb7v5t2q3u

Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

Marién Abreu, John Baptist Gauci, Domenico Labbate, Giuseppe Mazzuoccolo, Jean Paul Zerafa
2021 Electronic Journal of Combinatorics  
In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices,  ...  In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property.  ...  We will prove that L(G) is PMH in all of the following cases: • G is Hamiltonian with maximum degree ∆(G) at most 3 (Theorem 5), • G is a complete graph (Theorem 14), and • G is arbitrarily traceable from  ... 
doi:10.37236/9143 fatcat:lhlwzvs7s5gsvclsuixgnxftpq

Extending perfect matchings to Hamiltonian cycles in line graphs [article]

Marién Abreu, John Baptist Gauci, Domenico Labbate, Giuseppe Mazzuoccolo, Jean Paul Zerafa
2020 arXiv   pre-print
In particular, we prove that this happens when G is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, or (iii) an arbitrarily traceable graph.  ...  In this paper we establish some sufficient conditions for a graph G in order to guarantee that its line graph L(G) has the PMH-property.  ...  Figure 2 : 2 having maximum degree 4, and let M be the perfect matching of L(G) shown in the figure. A Hamiltonian graph with maximum degree 4 whose line graph is not PMH.  ... 
arXiv:1910.01553v2 fatcat:hlody5g6uzbjvpt7yufwu56zri

Locally self-avoiding Eulerian tours

Tien-Nam Le
2018 Journal of combinatorial theory. Series B (Print)  
It was independently conjectured by Häggkvist in 1989 and Kriesell in 2011 that given a positive integer , every simple eulerian graph with high minimum degree (depending on ) admits an eulerian tour such  ...  that every segment of length at most of the tour is a path.  ...  For the case = 3, i.e. triangle-free Eulerian tours, Adelgren [1] characterized all graphs with maximum degree at most 4 which admit a triangle-free Eulerian tour before Oksimets [12] proved Conjecture  ... 
doi:10.1016/j.jctb.2018.08.008 fatcat:5tczk3n7jrca7emeanjuaplsqq

The traveling salesman problem on cubic and subcubic graphs [article]

Sylvia Boyd, René Sitters, Suzanne van der Ster, Leen Stougie
2011 arXiv   pre-print
In fact we prove constructively that for any cubic graph on n vertices a tour of length 4n/3-2 exists, which also implies the 4/3 conjecture, as an upper bound, for this class of graph-TSP.  ...  We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest.  ...  maximum degree is 6.  ... 
arXiv:1107.1052v1 fatcat:sbklbt5rebaytnvbpsd7atmd7q

Locally self-avoiding eulerian tours [article]

Tien-Nam Le
2017 arXiv   pre-print
It was independently conjectured by Häggkvist in 1989 and Kriesell in 2011 that given a positive integer ℓ, every simple eulerian graph with high minimum degree (depending on ℓ) admits an eulerian tour  ...  such that every segment of length at most ℓ of the tour is a path.  ...  For the case ℓ = 3, i.e. triangle-free eulerian tours, Adelgren [1] characterized all graphs with maximum degree at most 4 which admit a triangle-free eulerian tour before Oksimets [12] proved Conjecture  ... 
arXiv:1611.07486v2 fatcat:iabobhglnfftpklfo4y5ry52im

Classes of line graphs with small cycle double covers

Gary MacGillivray, Karen Seyffarth
2001 The Australasian Journal of Combinatorics  
Bondy, states that every simple bridgeless graph on n vertices has a cycle double cover with at most (n -1) cycles.  ...  In this article, we prove that the conjecture holds for line graphs of a number of types of graphs; specifically line graphs of complete graphs, line graphs of complete bipartite graphs, and line graphs  ...  Then q-l Ee = U Pi, i=O is an eulerian subgraph of L( G) with maximum degree at most four.  ... 
dblp:journals/ajc/MacGillivrayS01 fatcat:b24uo33rt5cdxjl355rcnazc64

Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem

Hans-Joachim Böckenhauer, Juraj Hromkovič, Ralf Klasing, Sebastian Seibert, Walter Unger
2002 Theoretical Computer Science  
We show how to modify the Christoÿdes algorithm for -TSP to obtain e cient approximation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by  ...  The investigation of the possibility to e ciently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory.  ...  Using this we avoid taking edges with high costs, and the cost of our resulting graph G (and also of the corresponding Eulerian tour D) will be again (as in the case of -TSP) at most 3 2 times the cost  ... 
doi:10.1016/s0304-3975(01)00287-0 fatcat:xza5twvvg5gfzcnv2g4uypmr7i

Approximation algorithms for general cluster routing problem [article]

Xiaoyan Zhang, Donglei Du, Gregory Gutin, Qiaoxia Ming, Jian Sun
2020 arXiv   pre-print
The goal is to find a minimum cost walk T that visits each vertex in V' only once, traverses every edge in E' at least once and for every i∈ [k] all vertices of C_i are traversed consecutively.  ...  The weight function c satisfies the triangle inequality.  ...  Hence the length of the tour is at most 3 2 OP T − D + (3L − D) = 3 2 OP T + 3L − 2D. Proof of Theorem 8 Let T be the tour output by Algorithm 4.  ... 
arXiv:2006.12929v1 fatcat:f4cehmkkovh3lc3xoxeazicidy

The complexity of $P$4-decomposition of regular graphs and multigraphs

Ajit Diwan, Justine Dion, David Mendell, Michael Plantholt, Shailesh Tipnis
2015 Discrete Mathematics & Theoretical Computer Science  
International audience Let G denote a multigraph with edge set E(G), let µ(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices.  ...  Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3.  ...  Acknowledgements We would like to thank the anonymous referee for a careful reading of our paper and in particular for pointing out an error in our original proof of Theorem 11.  ... 
doi:10.46298/dmtcs.2128 fatcat:mnzr7byczjdlxmwxj6aee6isny
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