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T -preserving homomorphisms of oriented graphs

Jaroslav Nešetřil, Eric Sopena, Laurence Vignal (1997)

Commentationes Mathematicae Universitatis Carolinae

A homomorphism of an oriented graph G = ( V , A ) to an oriented graph G ' = ( V ' , A ' ) is a mapping ϕ from V to V ' such that ϕ ( u ) ϕ ( v ) is an arc in G ' whenever u v is an arc in G . A homomorphism of G to G ' is said to be T -preserving for some oriented graph T if for every connected subgraph H of G isomorphic to a subgraph of T , H is isomorphic to its homomorphic image in G ' . The T -preserving oriented chromatic number χ → T ( G ) of an oriented graph G is the minimum number of vertices in an oriented graph G ' such that there exists a T -preserving...

Teória grafov v chémii

Vladimír Baláž, Vladimír Kvasnička, Jiří Pospíchal (1991)

Pokroky matematiky, fyziky a astronomie

Tessellations of random maps of arbitrary genus

Grégory Miermont (2009)

Annales scientifiques de l'École Normale Supérieure

We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing one to encode such structures by labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these...

Testing Cayley graph densities

Goulnara N. Arzhantseva, Victor S. Guba, Martin Lustig, Jean-Philippe Préaux (2008)

Annales mathématiques Blaise Pascal

We present a computer-assisted analysis of combinatorial properties of the Cayley graphs of certain finitely generated groups: given a group with a finite set of generators, we study the density of the corresponding Cayley graph, that is, the least upper bound for the average vertex degree (= number of adjacent edges) of any finite subgraph. It is known that an m -generated group is amenable if and only if the density of the corresponding Cayley graph equals to 2 m . We test amenable and non-amenable...

Tetracyclic harmonic graphs

B. Borovićanin, I. Gutman, M. Petrović (2002)

Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques

Tetravalent Arc-Transitive Graphs of Order 3p 2

Mohsen Ghasemi (2014)

Discussiones Mathematicae Graph Theory

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 3p2 is given

Tetravalent half-arc-transitive graphs of order p 2 q 2

Hailin Liu, Bengong Lou, Bo Ling (2019)

Czechoslovak Mathematical Journal

We classify tetravalent G -half-arc-transitive graphs Γ of order p 2 q 2 , where G ≤ 𝖠𝗎𝗍 Γ and p , q are distinct odd primes. This result involves a subclass of tetravalent half-arc-transitive graphs of cube-free order.

The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs

Daniel W. Cranston, Sogol Jahanbekam, Douglas B. West (2014)

Discussiones Mathematicae Graph Theory

The 1, 2, 3-Conjecture states that the edges of a graph without isolated edges can be labeled from {1, 2, 3} so that the sums of labels at adjacent vertices are distinct. The 1, 2-Conjecture states that if vertices also receive labels and the vertex label is added to the sum of its incident edge labels, then adjacent vertices can be distinguished using only {1, 2}. We show that various configurations cannot occur in minimal counterexamples to these conjectures. Discharging then confirms the conjectures...

The 3-path-step operator on trees and unicyclic graphs

Bohdan Zelinka (2002)

Mathematica Bohemica

E. Prisner in his book Graph Dynamics defines the k -path-step operator on the class of finite graphs. The k -path-step operator (for a positive integer k ) is the operator S k ' which to every finite graph G assigns the graph S k ' ( G ) which has the same vertex set as G and in which two vertices are adjacent if and only if there exists a path of length k in G connecting them. In the paper the trees and the unicyclic graphs fixed in the operator S 3 ' are studied.

The 3-Rainbow Index of a Graph

Lily Chen, Xueliang Li, Kang Yang, Yan Zhao (2015)

Discussiones Mathematicae Graph Theory

Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ ℕ, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex subset S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for each k-subset S of V (G) is called the k-rainbow index of G, denoted by rxk(G). In this paper,...

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