scholarly journals A Note on the Number of Edges Guaranteeing a $C_4$ in Eulerian Bipartite Digraphs

10.37236/1667 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Jian Shen ◽  
Raphael Yuster
Keyword(s):  
Upper Bound ◽  
Directed Cycle ◽  
Type Result ◽  
Vertex Partition ◽  
Bipartite Digraph ◽  
Interchange Graphs

Let $G$ be an Eulerian bipartite digraph with vertex partition sizes $m,n$. We prove the following Turán-type result: If $e(G) > 2mn/3$ then $G$ contains a directed cycle of length at most 4. The result is sharp. We also show that if $e(G)=2mn/3$ and no directed cycle of length at most 4 exists, then $G$ must be biregular. We apply this result in order to obtain an improved upper bound for the diameter of interchange graphs.

scholarly journals Sparse Highly Connected Spanning Subgraphs in Dense Directed Graphs

2018 ◽  
Vol 28 (3) ◽  
pp. 423-464 ◽  
Author(s):  
DONG YEAP KANG
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Analogous Result ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Additional Term ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Polynomial Time Algorithms ◽  
Bipartite Digraph

Mader proved that every strongly k-connected n-vertex digraph contains a strongly k-connected spanning subgraph with at most 2kn - 2k2 edges, where equality holds for the complete bipartite digraph DKk,n-k. For dense strongly k-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly k-connected n-vertex digraph D contains a strongly k-connected spanning subgraph with at most kn + 800k(k + Δ(D)) edges, where Δ(D) denotes the maximum degree of the complement of the underlying undirected graph of a digraph D. Here, the additional term 800k(k + Δ(D)) is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly k-connected n-vertex semicomplete digraph contains a strongly k-connected spanning subgraph with at most kn + 800k2 edges, which is essentially optimal since 800k2 cannot be reduced to the number less than k(k - 1)/2.We also prove an analogous result for strongly k-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.

scholarly journals Nordhaus-Gaddum Type Results for Total Domination

2011 ◽  
Vol Vol. 13 no. 3 (Graph Theory) ◽  
Author(s):  
Michael Henning ◽  
Ernst Joubert ◽  
Justin Southey
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Domination Number ◽  
Total Domination ◽  
Type Result ◽  
Total Domination Number ◽  
Maximum Value ◽  
International Audience ◽  
Domination Numbers

Graph Theory International audience A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s,s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of G1 and G2 is 2s+4, with equality if and only if G1 = sK2 or G2 = sK2, while the maximum value of the product of the total domination numbers of G1 and G2 is max{8s,⌊(s+6)2/4 ⌋}.

scholarly journals NORDHAUS – GADDUM TYPE RESULTS FOR WIENER LIKE INDICES OF GRAPHS

2021 ◽  
Vol 58 (2) ◽  
pp. 5843-5854
Author(s):  
R. BHUVANESHWARI, V. KALADEVI
Keyword(s):  
Graph Theory ◽  
Regular Graph ◽  
Upper Bound ◽  
Wiener Index ◽  
Type Result

A Nordhaus - Gaddum type result is a lower or upper bound on sum or product of a parameter of a graph and its complement. This concept was introduced in 1956 by Nordhaus E. A.,             Gaddum J. W. Generalized Wiener like indices such as wiener index, Detour index, Reciprocal- wiener index, Harary- wiener index, Hyper- wiener index, Reciprocal- Detour index, Harary- Detour index and Hyper- Detour index have been studied in graph theory. In this paper, Nordhaus – Gaddum type results of these indices for k-Sun graph and four regular graph are presented.

scholarly journals On the minimum size of a point set containing a 5-hole and a disjoint 4-hole

2011 ◽  
Vol 48 (4) ◽  
pp. 445-457 ◽  
Author(s):  
Bhaswar Bhattacharya ◽  
Sandip Das
Keyword(s):  
Upper Bound ◽  
Minimum Size ◽  
Convex Hulls ◽  
Type Result ◽  
Convex Polygons ◽  
Empty Convex ◽  
Ramsey Type ◽  
Point Set

Let H(k; l), k ≦ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≦ H(4, 5) ≦ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.

Sign in / Sign up

Export Citation Format

Share Document