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### Identity orientation of complete bipartite graphs

Frank Harary, Desh Ranjan
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We show that the complete bipartite graph K s,t , with s t, does not have an identity orientation if t 3 s − log 3 (s −1) .  ...  An identity orientation of a graph G = (V , E) is an orientation of some of the edges of E such that the resulting partially oriented graph has no automorphism other than the identity.  ...  Harary and Jacobson [4] posed the following problem: For which values of s, t does the complete bipartite graph K s,t have an identity orientation?  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2004.07.017">doi:10.1016/j.disc.2004.07.017</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fbifhxumcvdbjhwauqdt64ssqq">fatcat:fbifhxumcvdbjhwauqdt64ssqq</a> </span>
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### Destroying symmetry by orienting edges: Complete graphs and complete bigraphs

Frank Harary, Michael S. Jacobson
<span title="">2001</span> <i title="Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g64zvtslcjdqncvrfbmvf422qi" style="color: black;">Discussiones Mathematicae Graph Theory</a> </i> &nbsp;
We find that this number for complete graphs is related to the number of identity oriented trees. For complete bipartite graphs K s,t , s ≤ t, this number does not always exist.  ...  Abstract Our purpose is to introduce the concept of determining the smallest number of edges of a graph which can be oriented so that the resulting mixed graph has the trivial automorphism group.  ...  Our object is to study the subtle problems of considering graphs which contain an io-set and of determining the values of the invariant io (G) for complete bipartite graphs and complete graphs.  ...
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### Distinguishing colorings of Cartesian products of complete graphs

Michael J. Fisher, Garth Isaak
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K s,t which admits only the identity automorphism.  ...  In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs.  ...  Acknowledgements: The authors would like to thank Peter Hammer for encouraging the writing of this paper. Garth Isaak would like to thank the Reidler Foundation for partial support of this research.  ...
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### Distinguishing colorings of Cartesian products of complete graphs [article]

Michael J. Fisher
<span title="2006-07-19">2006</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K_s,t which admits only the identity automorphism.  ...  In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs.  ...  Acknowledgements: The authors would like to thank Peter Hammer for encouraging the writing of this paper. Garth Isaak would like to thank the Reidler Foundation for partial support of this research.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0607465v1">arXiv:math/0607465v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zgqdbvx7azfxdcp33qapdo7see">fatcat:zgqdbvx7azfxdcp33qapdo7see</a> </span>
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### Distinguishing numbers and distinguishing indices of oriented graphs [article]

Kahina Meslem, Eric Sopena
<span title="2020-05-15">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we study the four corresponding parameters for oriented graphs whose underlying graph is a path, a cycle, a complete graph or a bipartite complete graph.  ...  In each case, we determine their minimum and maximum value, taken over all possible orientations of the corresponding underlying graph, except for the minimum values for unbalanced complete bipartite graphs  ...  Complete bipartite graphs: other cases We consider in this section the remaining cases of complete bipartite graphs, that is, unbalanced complete bipartite graphs K m,n with 2 ≤ m < n.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1910.12738v2">arXiv:1910.12738v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/w6f2qbbssvegllntl7vubqgl74">fatcat:w6f2qbbssvegllntl7vubqgl74</a> </span>
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### Eigenvalues of oriented-graph matrices

Jiong-Sheng Li
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
We also consider spectral properties of the bipartite oriented-graph matrices and the multiequipartite oriented-graph matrices.  ...  We give bounds on the real and imaginary parts of the eigenvalues of an oriented-graph matrix, prove that each of the irreducible oriented-graph matrices of order n 2 3 has at least three distinct eigenvalues  ...  BIPARTITE ORIENTED-GRAPH MATRICES In this section, we consider bounds on the spectral radius of the bipartite oriented-graph matrices.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0024-3795(93)00233-p">doi:10.1016/0024-3795(93)00233-p</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ad57nunjrzgcfjvu5kf7stnmby">fatcat:ad57nunjrzgcfjvu5kf7stnmby</a> </span>
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### Upper bounds on the numbers of 1-factors and 1-factorizations of hypergraphs

Anna Taranenko
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fhi2xwpnh5gmlgof2idwu5wlgq" style="color: black;">Electronic Notes in Discrete Mathematics</a> </i> &nbsp;
We estimate the number of 1-factors of uniform hypergraphs and the number of 1-factorizations of complete uniform hypergraphs by means of permanents of their adjacency matrices.  ...  A 1factorization of G is a partition of all hyperedges of the hypergraph into disjoint 1-factors.  ...  Also, permanents can be used for the estimation of the number of the complete graph 1-factorizations [7] . Let Φ(n) denote the number of 1-factorizations of the complete graph K n on n vertices.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2015.06.014">doi:10.1016/j.endm.2015.06.014</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2zcnem6oprgvbmyygfngwh7yfe">fatcat:2zcnem6oprgvbmyygfngwh7yfe</a> </span>
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### Orientation distance graphs revisited

<span title="">2007</span> <i title="Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g64zvtslcjdqncvrfbmvf422qi" style="color: black;">Discussiones Mathematicae Graph Theory</a> </i> &nbsp;
We provide new results about orientation distance graphs and simpler proofs to existing results, especially with regards to the bipartiteness of orientation distance graphs and the representation of orientation  ...  The orientation distance graph D o (G) of a graph G is defined as the graph whose vertex set is the pair-wise non-isomorphic orientations of G, and two orientations are adjacent iff the reversal of one  ...  We saw earlier (Theorem 1) that the orientation distance graphs of most complete bipartite graphs are bipartite. However, D o (K 3,3 ) is not bipartite.  ...
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### Delay Colouring in Quartic Graphs

Katherine Edwards, W. Sean Kennedy
<span title="2020-08-07">2020</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;
Haxell, Wilfong, and Winkler conjectured that every bipartite graph with maximum degree \$\Delta\$ is \$(\Delta + 1)\$-delay-colourable. We prove this conjecture in the special case \$\Delta = 4\$.  ...  Observe also that a transitive tournament is completely defined by its outdegrees, and hence a clique transitive orientation of a graph that is the edge disjoint union of cliques is completely defined  ...  Preliminaries In order to prove Theorem 2, it is enough to show it holds for 4-regular graphs, since every bipartite graph G is a subgraph of a ∆(G)-regular bipartite graph.  ...
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### Author index to volume 232 (2001)

<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Lewis, An algebraic identity of F.H. Jackson and its implications for partitions (Note) (1-3) 77} 83 Andrews, G.E. and R.P. Lewis, Restricted bipartitions (Note) (1-3) 85} 89 Bayat, M. and H.  ...  Ling, The existence of referee squares (Note) (1-3) 109}112 Fronc\ ek, D., Corrigendum to`Almost self-complementary factors of complete bipartite graphsa [Discrete Math. 167/168 (1997) 317}327] (1-3) 195  ...
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### Skew-spectra and skew energy of various products of graphs [article]

Xueliang Li, Huishu Lian
<span title="2013-05-31">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Moreover, we consider the skew energy of the orientation of the lexicographic product H[G] of a bipartite graph H and a graph G.  ...  In this paper, we give orientations of the Kronecker product H⊗ G and the strong product H∗ G of H and G where H is a bipartite graph and G is an arbitrary graph.  ...  The orientation of H * G Now we consider the strong product H * G of a bipartite graph H and a graph G, Let H τ be an oriented graph of H and G σ be an oriented graph of G.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1305.7305v1">arXiv:1305.7305v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ibwu3z25gngororem44fqiosh4">fatcat:ibwu3z25gngororem44fqiosh4</a> </span>
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### On the Turán number for the hexagon

Zoltan Füredi, Assaf Naor, Jacques Verstraëte
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/37jjomjmvfhrzf3di2gjnrrfuu" style="color: black;">Advances in Mathematics</a> </i> &nbsp;
A long-standing conjecture of Erdős and Simonovits is that ex(n, C 2k  ...  Recall that a maximal complete bipartite subgraph of G is a complete bipartite subgraph of G which contains a cycle and it not properly contained in any other complete bipartite subgraph of G.  ...  By Theorem 3.1, (G) is a decomposition of G into maximal complete bipartite graphs and single edges.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.aim.2005.04.011">doi:10.1016/j.aim.2005.04.011</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2v4o6cnb7vajngym3hg4kiryjy">fatcat:2v4o6cnb7vajngym3hg4kiryjy</a> </span>
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### The medial graph and voltage- current duality

Dan Archdeacon
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
., The medial graph and voltage-current duality, Discrete Mathematics 104 (1992) 111-141.  ...  vertices, and each edge by the complete bipartite graph K,,, on these new vertices.  ...  In general we would like to replace vertices of G with sets of independent vertices, and edges with complete bipartite graphs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(92)90328-d">doi:10.1016/0012-365x(92)90328-d</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5gycmsaj5vehvoe5vj4qfs67v4">fatcat:5gycmsaj5vehvoe5vj4qfs67v4</a> </span>
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### Simple eigenvalues of cubic vertex-transitive graphs [article]

Krystal Guo, Bojan Mohar
<span title="2020-02-13">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
If v is an eigenvector for eigenvalue λ of a graph X and α is an automorphism of X, then α(v) is also an eigenvector for λ.  ...  Thus it is rather exceptional for an eigenvalue of a vertex-transitive graph to be simple.  ...  We give several families of graphs with such spectral property, and completely classify some of special subfamilies.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.05694v1">arXiv:2002.05694v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nzo6jqaya5d3pknidys3is33iq">fatcat:nzo6jqaya5d3pknidys3is33iq</a> </span>
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### Antipodal covers of strongly regular graphs

Aleksandar Jurišić
<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;
This generalizes Drake's and Gardiner's characterization of distance-regular antipodal covers of complete bipartite graphs.  ...  Finally, antipodal covers of complete bipartite graphs and their line graphs are characterized in terms of weak resolvable transversal designs which are, in the case of maximal covering index, equivalent  ...  Antipodal covers of the lattice graphs and the complete bipartite graphs are characterized in Section 4.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(97)00139-8">doi:10.1016/s0012-365x(97)00139-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/shuni5ozqffhroi3psagupjgyy">fatcat:shuni5ozqffhroi3psagupjgyy</a> </span>
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