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Hajós' conjecture and small cycle double covers of planar graphs

Karen Seyffarth
1992 Discrete Mathematics  
., Hajos' conjecture and small cycle double covers of planar graphs, Discrete Mathematics 101 (1992) 291-306.  ...  We also discuss the connection between this result and the Small Cycle Double Cover Conjecture.  ...  Acknowledgements Work on this paper was supported in part by the Natural Sciences and Engineering Research Council of Canada. Thanks are due to the referees for their helpful remarks.  ... 
doi:10.1016/0012-365x(92)90610-r fatcat:f7ckgeyznzhx3l4n3uca3lst4q

Small Oriented Cycle Double Cover of Graphs

Behrooz Bagheri Gh., Behnaz Omoomi
2015 Bulletin of the Malaysian Mathematical Sciences Society  
Keywords: Cycle double cover, Small cycle double cover, Oriented cycle double cover, Small oriented cycle double cover.  ...  A small oriented cycle double cover (SOCDC) of a bridgeless graph G on n vertices is a collection of at most n −1 directed cycles of the symmetric orientation, G s , of G such that each arc of G s lies  ...  Zhang for reading the manuscript and for his helpful suggestions.  ... 
doi:10.1007/s40840-015-0169-2 fatcat:rbywcpzgo5g5njzniqmvofg4pq

Small oriented cycle double cover of graphs [article]

Behrooz Bagheri Gh., Behnaz Omoomi
2013 arXiv   pre-print
A small oriented cycle double cover (SOCDC) of a bridgeless graph G on n vertices is a collection of at most n-1 directed cycles of the symmetric orientation, G_s, of G such that each edge of G_s lies  ...  It is conjectured that every 2-connected graph except two complete graphs K_4 and K_6 has an SOCDC.  ...  A small cycle double cover (SCDC) of a graph on n vertices is a CDC with at most n − 1 cycles.  ... 
arXiv:1207.5122v2 fatcat:s76pgdu35reqbhwvka2yc6aqae

Page 1219 of Mathematical Reviews Vol. , Issue 93c [page]

1993 Mathematical Reviews  
These conjectures are closely related to the cycle double cover conjecture and the Hajos conjecture on decomposing even graphs into cycles.  ...  Let cc(G) and sc(G) denote the minimum number of cycles in a cycle cover and in a cycle double cover of a graph G, respectively. Bondy posed the following conjectures.  ... 

Cycle decompositions of pathwidth-6 graphs [article]

Elke Fuchs, Laura Gellert, Irene Heinrich
2017 arXiv   pre-print
This implies that these graphs satisfy the small cycle double cover conjecture.  ...  With these techniques we find structures that cannot occur in a minimal counterexample to Haj\'os conjecture and verify the conjecture for Eulerian graphs of pathwidth at most 6.  ...  Hajós' conjecture implies a conjecture of Bondy regarding the Cycle double cover conjecture. Conjecture 4 (Small Cycle Double Cover Conjecture (Bondy [3] )).  ... 
arXiv:1705.07066v2 fatcat:knxu5gn42belra4zows4psl6qm

Protection cycles in mesh WDM networks

G. Ellinas, A.G. Hailemariam, T.E. Stern
2000 IEEE Journal on Selected Areas in Communications  
It is also independent of the topology and the connection state of the network at the time of the failure.  ...  A fault recovery system that is fast and reliable is essential to today's networks, as it can be used to minimize the impact of the fault on the operation of the network and the services it provides.  ...  The short cycle double cover (SCDC) conjecture [31] states that every 2-edge connected simple graph of order has a CDC with fewer than cycles (holds for all maximal and Hamiltonian planar graphs).  ... 
doi:10.1109/49.887913 fatcat:grjiapis6japzl73c55vhmw3qe

Three ways to cover a graph [article]

Kolja Knauer, Torsten Ueckerdt
2015 arXiv   pre-print
As input graphs we consider graphs of bounded degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as well as outerplanar, planar bipartite, and planar graphs.  ...  Whereas the folded covering number has been investigated thoroughly for some covering classes, e.g., interval graphs and planar graphs, the local covering number has received little attention.  ...  Kolja Knauer was partially supported by DFG grant FE-340/8-1 as part of ESF project GraDR EUROGIGA and PEPS grant EROS and Torsten Ueckerdt by GraDR EUROGIGA project No. GIG/11/E023.  ... 
arXiv:1205.1627v3 fatcat:clp6x5vpb5cprcoz34zhcz5zsy

Some Recent Progress and Applications in Graph Minor Theory

Ken-ichi Kawarabayashi, Bojan Mohar
2007 Graphs and Combinatorics  
In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the "rough" structure of graphs excluding a fixed minor.  ...  This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation.  ...  Using similar technique, Kühn and Osthus also proved that Hajós' conjecture (which involves topological complete graph minors) is true for graphs of girth at least 186. See [101] .  ... 
doi:10.1007/s00373-006-0684-x fatcat:wkf3w6cemzc4hnnw3neqpck6hu

On measures of edge-uncolorability of cubic graphs: A brief survey and some new results [article]

M.A. Fiol, G. Mazzuoccolo, E. Steffen
2017 arXiv   pre-print
There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs.  ...  The paper closes with a list of open problems and conjectures.  ...  The research of the third author on this project was supported by Deutsche Forschungsgemeinschaft (DFG) grant STE 792/2-1.  ... 
arXiv:1702.07156v1 fatcat:56e6ysreure75gupsvey2nuu5m

Graph colorings with local constraints -- a survey

Zsolt Tuza
1997 Discussiones Mathematicae Graph Theory  
In one of the most general formulations, a graph G = (V, E), sets L(v) of admissible colors, and natural numbers c v for the vertices v ∈ V are given, and the question is whether there can be chosen a  ...  The particular case of constant |L(v)| with c v = 1 for all v ∈ V leads to the concept of choice number, a graph parameter showing unexpectedly different behavior compared to the chromatic number, despite  ...  that covers the smaller vertex class of a bipartite graph).  ... 
doi:10.7151/dmgt.1049 fatcat:du2wuuygubedxp5jzztrmw5ike

Measures of Edge-Uncolorability of Cubic Graphs

M. A. Fiol, G. Mazzuoccolo, E. Steffen
2018 Electronic Journal of Combinatorics  
There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the $5$-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs.  ...  The paper closes with a list of open problems and conjectures.  ...  Acknowledgements The authors sincerely acknowledge the useful comments and suggestions of the reviewers, which led to a significant improvement of this work.  ... 
doi:10.37236/6848 fatcat:gp6i46upuzaonfaw3qxfrrg7zm

Subject Index Volumes 1–200

2001 Discrete Mathematics  
4418 Häggkvist Mitchem Schmeichel, 1563 Hahn, 1729 Hajós, 3273 Halin, 3127 Harary, 3943, 4880, 5859 Harary Robinson Wormald, 4395 Hartman, 4237, 4529, 5479 Heawood, 5823 Hedetniemi, 4656 Hedietniemi  ...  , 5765 Henning and Slater, 5299 Hertz and de Werra, 3748 Hibi, 4839 Hoàng, 3748, 4811 Holton, 1660 Holyer and Cockayne, 1118 Howorka, 2636  ...  arc, 5385 cap, 5394 small cut set, 4822 cycle, 3087 small cycle double cover, 2512, 3273, see also SCDC conjecture, 3273 small degree, 4384, 5953 covering, 4153, 4216 unramified covering, 4153  ... 
doi:10.1016/s0012-365x(00)00471-4 fatcat:znxg54q6ifgutnhzj3urcw455y

SOME APPLICATIONS OF GRAPH THEORY

FRED ROBERTS
2001 Combinatorial and Computational Mathematics  
For each, we mention a variety of applications, discuss generalizations of the concept arising from the applications, and describe some recent results and open questions.  ...  We review three graph-theoretical concepts: graph coloring, intersection graph, and competition graph.  ...  For example, Erdös, Rubin, and Taylor [32] conjectured that every planar graph is 5-choosable but that there are planar graphs that are not 4-choosable.  ... 
doi:10.1142/9789812799890_0008 fatcat:jxwgnebbgfdzzcb5ewa3gnwkku

Structure of classes of graphs defined by forbidding induced subgraphs [article]

Nicolas Trotignon
2013 arXiv   pre-print
This document is the Habilitation Thesis (Habilitation \'a Diriger des Recherches) of Nicolas Trotignon. It presents his work in structural graph theory, from 2001 to 2009.  ...  Adding path-cobipartite graphs and path-double split graphs as basic graphs in our conjecture is not enough.  ...  Also, ISK4-free graphs generalize series-parallel graphs, so bounding their chromatic number would be a kind of generalisation of Hajos and Hadwiger conjectures which are the same easy statements for K  ... 
arXiv:1308.6678v1 fatcat:xgtoz7wi5nd7df3efa3lanoidq

Page 178 of Mathematical Reviews Vol. , Issue Subject Index [page]

Mathematical Reviews  
(English summary) 2002g:68153 Huck, Andreas On cycle-double covers of graphs of small oddness.  ...  C. see Fischer, Ise, 2002d:05103 Liu, Tong-yin (with Liu, Yan Pei) Equilibrium small circuit double covers of near- triangulations.  ... 
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