scholarly journals Hereditary biclique-Helly graphs: recognition and maximal biclique enumeration

2013 ◽  
Vol Vol. 15 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Martiniano Eguia ◽  
Francisco Juan Soulignac
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Induced Subgraph ◽  
Helly Property ◽  
International Audience

Graphs and Algorithms International audience A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C4-dominated graphs that contain no triangles and no induced cycles of length either 5 or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n2+αm) time and O(n+m) space. (Here n, m, and α= O(m1/2) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C4-dominated graphs that contain no triangles in O(αm) time and O(n+m) space. Finally, we show how to enumerate all the maximal bicliques of a C4-dominated graph with no triangles in O(n2 + αm) time and O(αm) space.

scholarly journals On hereditary Helly classes of graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Marina Groshaus ◽  
Jayme Luiz Szwarcfiter
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Natural Question ◽  
Fixed Size ◽  
Forbidden Subgraphs ◽  
Induced Subgraph ◽  
Helly Property ◽  
International Audience

Graphs and Algorithms International audience In graph theory, the Helly property has been applied to families of sets, such as cliques, disks, bicliques, and neighbourhoods, leading to the classes of clique-Helly, disk-Helly, biclique-Helly, neighbourhood-Helly graphs, respectively. A natural question is to determine for which graphs the corresponding Helly property holds, for every induced subgraph. This leads to the corresponding classes of hereditary clique-Helly, hereditary disk-Helly, hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. In this paper, we describe characterizations in terms of families of forbidden subgraphs, for the classes of hereditary biclique-Helly and hereditary neighbourhood-Helly graphs. We consider both open and closed neighbourhoods. The forbidden subgraphs are all of fixed size, implying polynomial time recognition for these classes.

scholarly journals Computing the number of h-edge spanning forests in complete bipartite graphs

2014 ◽  
Vol Vol. 16 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Rebecca Stones
Keyword(s):  
Bipartite Graph ◽  
Spanning Trees ◽  
Analysis Of Algorithms ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Spanning Forests ◽  
International Audience ◽  
Complete Bipartite Graphs ◽  
Number Of Spanning Trees

Analysis of Algorithms International audience Let fm,n,h be the number of spanning forests with h edges in the complete bipartite graph Km,n. Kirchhoff\textquoterights Matrix Tree Theorem implies fm,n,m+n-1=mn-1 nm-1 when m ≥1 and n ≥1, since fm,n,m+n-1 is the number of spanning trees in Km,n. In this paper, we give an algorithm for computing fm,n,h for general m,n,h. We implement this algorithm and use it to compute all non-zero fm,n,h when m ≤50 and n ≤50 in under 2 days.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

Generalization of the Symmetric Starter of the Orthogonal Double Covers of the Complete Bipartite Graph

2006 ◽  
Vol 16 (2) ◽  
pp. 171-177
Author(s):  
S. A. El-Serafi ◽  
R. A. El-Shanawany ◽  
M. Sh. Higazy
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Double Covers

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

Spanning Trees of the Complete Bipartite Graph

1990 ◽  
pp. 339-346 ◽  
Author(s):  
N. Hartsfield ◽  
J. S. Werth
Keyword(s):  
Bipartite Graph ◽  
Spanning Trees ◽  
Complete Bipartite Graph

scholarly journals An Exact Turán Result for Tripartite 3-Graphs

10.37236/5203 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Sanitt ◽  
John Talbot
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Maximum Size ◽  
Unique Graph

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollobás that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.

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