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

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*. ...

##
###
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

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, ...

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

2005
*
Operations Research Letters
*

Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP

doi:10.1016/j.orl.2004.09.005
fatcat:fnzbstj7qbby5l6fjzyuzixeqy
*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. ...##
###
TSP on Cubic and Subcubic Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

Using polyhedral techniques

doi:10.1007/978-3-642-20807-2_6
fatcat:donyaxftkfcmpd44bbthuyueay
*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. ...##
###
The traveling salesman problem on cubic and subcubic graphs

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. ...

##
###
Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

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 ...

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

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. ...

##
###
Locally self-avoiding Eulerian tours

2018
*
Journal of combinatorial theory. Series B (Print)
*

It was independently conjectured by Häggkvist

doi:10.1016/j.jctb.2018.08.008
fatcat:5tczk3n7jrca7emeanjuaplsqq
*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 ...##
###
The traveling salesman problem on cubic and subcubic graphs
[article]

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. ...

##
###
Locally self-avoiding eulerian tours
[article]

2017
*
arXiv
*
pre-print

It was independently conjectured by Häggkvist

arXiv:1611.07486v2
fatcat:iabobhglnfftpklfo4y5ry52im
*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 ...##
###
Classes of line graphs with small cycle double covers

2001
*
The Australasian Journal of Combinatorics
*

Bondy, states that every simple bridgeless

dblp:journals/ajc/MacGillivrayS01
fatcat:b24uo33rt5cdxjl355rcnazc64
*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. ...##
###
Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem

2002
*
Theoretical Computer Science
*

We show how to modify the Christoÿdes algorithm for -TSP to obtain e cient approximation algorithms

doi:10.1016/s0304-3975(01)00287-0
fatcat:xza5twvvg5gfzcnv2g4uypmr7i
*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 ...##
###
Approximation algorithms for general cluster routing problem
[article]

2020
*
arXiv
*
pre-print

The goal is to find a minimum cost walk T that visits each vertex

arXiv:2006.12929v1
fatcat:f4cehmkkovh3lc3xoxeazicidy
*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*. ...##
###
The complexity of $P$4-decomposition of regular graphs and multigraphs

2015
*
Discrete Mathematics & Theoretical Computer Science
*

International audience Let G denote a multigraph

doi:10.46298/dmtcs.2128
fatcat:mnzr7byczjdlxmwxj6aee6isny
*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. ...
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