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### A note on minimal directed graphs with given girth

Yahya Ould Hamidoune
<span title="">1987</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
Caccetta and HHggkvist conjectured that the minimum order of a directed graph with girth g and outdegree not less than r is r(g -1) + 1. They proved this conjecture for r=2. We prove it for r= 3.  ...  A. Bondy for many helpful suggestions and comments.  ...  The girth of a directed graph G having at least one circuit is the minimal length of a circuit of G. We denote it by g(G). Behzad et al.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0095-8956(87)90009-8">doi:10.1016/0095-8956(87)90009-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/55kasozutnfhddtfkxrwr6fk4u">fatcat:55kasozutnfhddtfkxrwr6fk4u</a> </span>
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### Graphs and digraphs with given girth and connectivity

Jiping Liu,, Huishan Zhou
<span title="">1994</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In this paper, we show that for any given two positive integers g and k with g > 3, there exists a graph (digraph) G with girth g and connectivity k.  ...  Applying this result, we give a negative answer to the problem proposed by M. Junger, G. Reinelt and W.R Pulleyblank (1985).  ...  There exists a (k, g)-cage [7] which is the minimal graph with girth g and regularity k; there also exists a regular graph with given girth pair [3] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(92)00576-d">doi:10.1016/0012-365x(92)00576-d</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q5bng32ybzaozap2crxrcfeke4">fatcat:q5bng32ybzaozap2crxrcfeke4</a> </span>
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### Oriented colorings of partial 2-trees

Pascal Ochem, Alexandre Pinlou
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5xfy4fcdxvcvdiybp6iqbffqc4" style="color: black;">Information Processing Letters</a> </i> &nbsp;
A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping f from is an arc in H whenever xy is an arc in G.  ...  The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H.  ...  Some authors then studied the oriented chromatic number of planar graphs with given girth to get some hints on the behaviour of this invariant.  ...
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### Codes for distributed storage from 3-regular graphs [article]

Shuhong Gao, Fiona Knoll, Felice Manganiello, Gretchen Matthews
<span title="2016-09-30">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This paper considers distributed storage systems (DSSs) from a graph theoretic perspective. A DSS is constructed by means of the path decomposition of a 3- regular graph into P4 paths.  ...  The paths represent the disks of the DSS and the edges of the graph act as the blocks of storage. We deduce the properties of the DSS from a related graph and show their optimality.  ...  Recall that given a connected graph G with m edges on n vertices with a girth of g, the code C E (G) has rate m−n+1 m = 1 − n−1 m .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1610.00043v1">arXiv:1610.00043v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nzd5of2jf5hazizrnfe2t22x2a">fatcat:nzd5of2jf5hazizrnfe2t22x2a</a> </span>
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### Constructions for Cubic Graphs with Large Girth

Norman Biggs
<span title="1998-08-31">1998</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
There are also some historical notes on the current-best graphs for girth up to 36.  ...  The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth.  ...  Girth 18 Bray, Parker, and Rowley [13] have constructed a graph with 3024 vertices. It is a Cayley graph of the direct product of PSL(2, 8) with a cyclic group of order 6.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/1386">doi:10.37236/1386</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2ttjoutfcbbmzcd6hch3ozych4">fatcat:2ttjoutfcbbmzcd6hch3ozych4</a> </span>
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### Constrained Representations of Map Graphs and Half-Squares

Hoang-Oanh Le, Van Bang Le, Michael Wagner
<span title="2019-08-23">2019</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5b35w7jagjckfngb6pj76nbam4" style="color: black;">International Symposium on Mathematical Foundations of Computer Science</a> </i> &nbsp;
has a witness with at most 3|V | − 6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete.  ...  Our proof relies on a characterization of half-squares of tree-convex bipartite graphs, saying that these are precisely the chordal and dually chordal graphs.  ...  One direction in attacking this problem is to consider map graphs with restricted witness.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.mfcs.2019.13">doi:10.4230/lipics.mfcs.2019.13</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/mfcs/LeL19.html">dblp:conf/mfcs/LeL19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lyte7e2k7rhc5pfgfqfdmkz7py">fatcat:lyte7e2k7rhc5pfgfqfdmkz7py</a> </span>
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### Embedding cycles of given length in oriented graphs

Daniela Kühn, Deryk Osthus, Diana Piguet
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
V(G)|/k\rfloor +1 contains a directed cycle of length l.  ...  Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |  ...  Acknowledgement The authors would like to thank the referees for a careful reading of the manuscript and for pointing out an oversight in our proof.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2012.10.002">doi:10.1016/j.ejc.2012.10.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/esmmvwn7mveqfkqukz43zbvdsu">fatcat:esmmvwn7mveqfkqukz43zbvdsu</a> </span>
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### On universal graphs for planar oriented graphs of a given girth

O.V. Borodin, A.V. Kostochka, J. Nešetřil, A. Raspaud, E. Sopena
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
If each graph in a class ~ has a homomorphism to the same H', then H' is ~-universal. Let ~k denote the class of orientations of planar graphs with girth at least k. Clearly, ~3 ~ ~4 ~ ~5...  ...  We discuss the existence of ~k-universal graphs with special properties. It is known (see Raspaud and Sopena, 1994 ) that there exists a ~3-universal graph on 80 vertices.  ...  There exists a planar graph on 6 vertices which is universal for the set of graphs in ,~/'~/~16/7 with girth at least 11. Note that, by (1), Theorem 2 yields the following.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(97)00276-8">doi:10.1016/s0012-365x(97)00276-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7rmeiloc3jcsvj5egmtj535iqa">fatcat:7rmeiloc3jcsvj5egmtj535iqa</a> </span>
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### Amalgams of Cubic Bipartite Graphs

Domenico Labbate
<span title="">2004</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c45m6ttnaje4xbjsq7m2c6df2a" style="color: black;">Designs, Codes and Cryptography</a> </i> &nbsp;
We define a "graph amalgam" to point out a different approach to certain classes of cubic bipartite graphs.  ...  The amalgamation technique has been introduced for groups by Higman et al. [8] and Goldschmidt [7] and developed on geometries by Kegel and Schleiermacher [9].  ...  G is a repeated star product of G 1 ; . . . ; G n if and only if G is a graph amalgam on a directed tree. Note.  ...
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### Well-indumatched Trees and Graphs of Bounded Girth [article]

S. Akbari, T. Ekim, A.H. Ghodrati, S. Zare
<span title="2019-12-16">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
graph of girth g.  ...  On the other hand, there are infinitely many well-indumatched unicyclic graphs of girth k, where k is in 3, 5, 7 or k is an even integer greater than 2.  ...  It follows from Theorem 2 that one can decide in time O(m) whether a given graph is minimal well-indumatched of girth at least 9.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1903.03197v2">arXiv:1903.03197v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uqxica22f5fmzminvnjjk7za5y">fatcat:uqxica22f5fmzminvnjjk7za5y</a> </span>
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### Colorings and girth of oriented planar graphs

Jarik Nešetřil, André Raspaud, Eric Sopena
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In particular, we prove that every orientation of any large girth planar graph is Scolorable and classify those digraphs on 3, 4 and 5 vertices which color all large girth oriented planar graphs.  ...  Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number.  ...  One last note regarding oriented coloring of graphs with large girth. Let G = (V, E) be an undirected graph.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(96)00198-7">doi:10.1016/s0012-365x(96)00198-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jue4vhi7tfeppdyyg7qe5hu6ra">fatcat:jue4vhi7tfeppdyyg7qe5hu6ra</a> </span>
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### Page 1717 of Mathematical Reviews Vol. , Issue 2004c [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
” 2004¢:05100 05C35 05C40 Yuster, Raphael (IL-HAIF2; Tivon) A note on graphs with k-connected subgraphs.  ...  Given a route system one can define the underlying graph. If the route system satisfies a symmetry axiom, then the underlying graph is undirected; otherwise the underlying graph is directed.  ...
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### Map graphs having witnesses of large girth

Hoang-Oanh Le, Van Bang Le
<span title="">2019</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
In this note, we give good characterizations and efficient recognition for half-squares of bipartite graphs with girth at least a given integer g> 8.  ...  A half-square of a bipartite graph B=(X,Y,E_B) has one color class of B as vertex set, say X; two vertices are adjacent whenever they have a common neighbor in Y.  ...  One direction in attacking this problem is to consider map graphs with restricted witness.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2018.12.010">doi:10.1016/j.tcs.2018.12.010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/46ytm4krnvbnhapgwwjdh2ifjm">fatcat:46ytm4krnvbnhapgwwjdh2ifjm</a> </span>
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### Girth, words and diameter

Martin W. Liebeck, Aner Shalev
<span title="2019-04-10">2019</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/djmawcsftbaaxk2nkcppkone3i" style="color: black;">Bulletin of the London Mathematical Society</a> </i> &nbsp;
We study the girth of Cayley graphs of finite classical groups G on random sets of generators.  ...  We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.  ...  Introduction The girth of a graph (resp. directed graph) is the minimal length of a cycle (resp. directed cycle) in the graph.  ...
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200710033508/https://spiral.imperial.ac.uk/bitstream/10044/1/68007/2/girth7.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a4/96/a496c786a1b63ab413f7ce5a30e6d974b6687cf6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1112/blms.12251"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

### 6-cycle double covers of cubic graphs [article]

Rodrigo S. C. Leao, Valmir C. Barbosa
<span title="2009-04-17">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In [9], a particular instance of sense of direction, called a chordal sense of direction (CSD), is studied and the class of k-regular graphs that admit a CSD with exactly k labels (a minimal CSD) is analyzed  ...  A sense of direction is an edge labeling on graphs that follows a globally consistent scheme and is known to considerably reduce the complexity of several distributed problems.  ...  Acknowledgments The authors acknowledge partial support from CNPq, CAPES, and a FAPERJ BBP grant. They also thank C. Thomassen for pointing references [1, 14, 15] to them.  ...
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