scholarly journals Comparable Axiomatizations of the Myerson Value, the Restricted Banzhaf Value, Hierarchical Outcomes and the Average Tree Solution for Cycle-Free Graph Restricted Games

2009 ◽  
Author(s):  
René van den Brink
Keyword(s):  
Banzhaf Value ◽  
Free Graph ◽  
Myerson Value ◽  
Restricted Games

scholarly journals Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

scholarly journals Universal and unavoidable graphs

2021 ◽  
pp. 1-14
Author(s):  
Matija Bucić ◽  
Nemanja Draganić ◽  
Benny Sudakov
Keyword(s):  
Free Graph ◽  
Turán Number ◽  
Work Done

Abstract The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

scholarly journals The average tree solution for cycle-free graph games

2008 ◽  
Vol 62 (1) ◽  
pp. 77-92 ◽  
Author(s):  
P. Jean Jacques Herings ◽  
Gerard van der Laan ◽  
Dolf Talman
Keyword(s):  
Free Graph ◽  
Graph Games

Every 3-connected {K 1,3,N 3,3,3}-free graph is Hamiltonian

2013 ◽  
Vol 56 (8) ◽  
pp. 1585-1595 ◽  
Author(s):  
HouYuan Lin ◽  
ZhiQuan Hu
Keyword(s):  
Free Graph

Mixed Ramsey Numbers Revisited

2003 ◽  
Vol 12 (5-6) ◽  
pp. 653-660
Author(s):  
C. C. Rousseau ◽  
S. E. Speed
Keyword(s):  
Asymptotic Behaviour ◽  
Ramsey Number ◽  
Open Problems ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Dense Graphs

Given a graph Hwith no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with H fixed and , (ii) with m fixed and a sequence of dense graphs, in particular for the sequence . Open problems are mentioned throughout the paper.

scholarly journals On degree sums of a triangle-free graph

2014 ◽  
Vol 337 ◽  
pp. 76-82 ◽  
Author(s):  
Stephan Brandt ◽  
Jochen Harant ◽  
Steffi Naumann
Keyword(s):  
Free Graph

scholarly journals Context-free graph grammars

1978 ◽  
Vol 37 (2) ◽  
pp. 207-233 ◽  
Author(s):  
Pierluigi Della Vigna ◽  
Carlo Ghezzi
Keyword(s):  
Graph Grammars ◽  
Free Graph ◽  
Context Free

scholarly journals Graphs on Affine and Linear Spaces and Deuber Sets

10.37236/2732 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
David S. Gunderson ◽  
Hanno Lefmann
Keyword(s):  
Abelian Group ◽  
Finite Field ◽  
Recent Work ◽  
Independent Set ◽  
Large Set ◽  
Free Graph ◽  
Linear Spaces ◽  
Ramsey’S Theorem ◽  
Positive Integers ◽  
Arbitrary Abelian Group

If $G$ is a large $K_k$-free graph, by Ramsey's theorem, a large set of vertices is independent. For graphs whose vertices are positive integers, much recent work has been done to identify what arithmetic structure is possible in an independent set. This paper addresses  similar problems: for graphs whose vertices are affine or linear spaces over a finite field,  and when the vertices of the graph are elements of an arbitrary Abelian group.

scholarly journals An Ordered Turán Problem for Bipartite Graphs

10.37236/2471 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Craig Timmons
Keyword(s):  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Free Graph ◽  
The Family ◽  
Turan Problem ◽  
Vertex Graph ◽  
Turán Number ◽  
Turán Problem ◽  
Natural Way

Let $F$ be a graph.  A graph $G$ is $F$-free if it does not contain $F$ as a subgraph.  The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices.  The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem.  In this paper we introduce an ordered version of the Turán problem for bipartite graphs.  Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way.  A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$.  A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part.  Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles.  We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$.  In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$.  For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.

Sign in / Sign up

Export Citation Format

Share Document