scholarly journals Largest cliques in connected supermagic graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Anna Lladó
Keyword(s):  
Complete Graph ◽  
Additional Condition ◽  
Induced Subgraph ◽  
Sidon Sets ◽  
International Audience ◽  
Integer Labeling

International audience A graph $G=(V,E)$ is said to be $\textit{magic}$ if there exists an integer labeling $f: V \cup E \to [1, |V \cup E|]$ such that $f(x)+f(y)+f(xy)$ is constant for all edges $xy \in E$. Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most $3n^2+o(n^2)$ which contain a complete graph of order $n$. Bounds on Sidon sets show that the order of such a graph is at least $n^2+o(n^2)$. We close the gap between those two bounds by showing that, for any given graph $H$ of order $n$, there are connected magic graphs of order $n^2+o(n^2)$ containing $H$ as an induced subgraph. Moreover it can be required that the graph admits a supermagic labelling $f$, which satisfies the additional condition $f(V)=[1,|V|]$.

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Satisfy Condition ◽  
Induced Subgraph ◽  
Zero Divisor Graph ◽  
Ring Graph ◽  
Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals Random assignment and shortest path problems

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Johan Wästlund
Keyword(s):  
Limit Theorem ◽  
Shortest Path ◽  
Complete Graph ◽  
Assignment Problem ◽  
Shortest Path Problem ◽  
Random Assignment ◽  
Shortest Path Problems ◽  
Parisi Formula ◽  
International Audience ◽  
Random Assignment Problem

International audience We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.

scholarly journals Infinite limits and folding

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Anthony Bonato ◽  
Jeannette Janssen
Keyword(s):  
Biological Networks ◽  
Infinite Graphs ◽  
Isomorphism Type ◽  
Connected Component ◽  
Induced Subgraph ◽  
Pursuit Games ◽  
Graph Homomorphisms ◽  
International Audience

International audience We study infinite limits of graphs generated by the duplication model for biological networks. We prove that with probability 1, the sole nontrivial connected component of the limits is unique up to isomorphism. We describe certain infinite deterministic graphs which arise naturally from the model. We characterize the isomorphism type and induced subgraph structure of these infinite graphs using the notion of dismantlability from the theory of vertex pursuit games, and graph homomorphisms.

scholarly journals Parameterized Problems Related to Seidel's Switching

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Eva Jelinkova ◽  
Ondrej Suchy ◽  
Petr Hlineny ◽  
Jan Kratochvil
Keyword(s):  
Minimum Degree ◽  
The Other ◽  
Complete Graphs ◽  
Maximum Likelihood Decoding ◽  
Induced Subgraph ◽  
Fixed Parameter Tractable ◽  
Graph Theoretic ◽  
Np Completeness ◽  
Fixed Parameter ◽  
International Audience

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.

scholarly journals Long cycles in hypercubes with distant faulty vertices

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Linear Code ◽  
Minimum Distance ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
International Audience ◽  
Long Cycles

Graphs and Algorithms International audience In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.

scholarly journals Avoider-enforcer star games

2015 ◽  
Vol Vol. 17 no. 1 (Combinatorics) ◽  
Author(s):  
Andrzej Grzesik ◽  
Mirjana Mikalački ◽  
Zoltán Lóránt Nagy ◽  
Alon Naor ◽  
Balázs Patkós ◽  
...  
Keyword(s):  
Complete Graph ◽  
Order Of Magnitude ◽  
Winning Strategies ◽  
International Audience

Combinatorics International audience In this paper, we study (1 : b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k≥3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We consider both versions of Avoider-Enforcer games — the strict and the monotone — and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases fmonF, f-F and f+F, where F is the hypergraph of the game.

scholarly journals Cayley and Tutte polytopes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Matjaž Konvalinka ◽  
Igor Pak
Keyword(s):  
Complete Graph ◽  
Tutte Polynomial ◽  
Convex Hulls ◽  
Connected Graphs ◽  
Labeled Trees ◽  
Graph Traversal ◽  
International Audience ◽  
Search Graph ◽  
Traversal Algorithm

International audience Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to a two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation of the Cayley and Tutte polytope. We prove that simplices in the triangulations correspond to labeled trees and forests. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm. Les polytopes de Cayley ont été définis récemment comme des ensembles convexes de compositions de Cayley introduits par Cayley en 1857. Dans ce papier, nous résolvons la conjecture de Braun. Cette dernière exprime le volume du polytopes de Cayley en termes du nombre de graphes connexes. Nous étendons ce résultat à des déformations de polytopes de Cayley à deux variables, à savoir les polytopes de Tutte. Le volume de ces derniers est donnè par une évaluation du polynôme de Tutte du graphe complet. Notre approche est basée sur une triangulation explicite des polytopes de Cayley et Tutte. Nous démontrons que les simplexes de ces triangulations correspondent à des arbres marqués. La pierre angulaire de notre démonstration est une bijection directe basées sur l'algorithme de la recherche du premier voisin sur le graphe.

scholarly journals The analysis of a prioritised probabilistic algorithm to find large induced forests in regular graphs with large girth

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Carlos Hoppen
Keyword(s):  
Lower Bounds ◽  
Regular Graphs ◽  
Probabilistic Algorithms ◽  
Induced Subgraph ◽  
Fixed Integer ◽  
Asymptotic Bounds ◽  
Random Regular Graph ◽  
International Audience ◽  
Large Girth ◽  
Deterministic Bounds

International audience The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.

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