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General numeration II. Division schemes.

D. W. Dubois (1982)

Revista Matemática Hispanoamericana

This is the second in a series of two papers on numeration schemes. Whereas the first paper emphasized grouping as exemplified in the partition of a number so as to obtain its base two numeral, the present paper takes at its point of departure the method of repeated divisions, as in the calculation of the base two numeral for a number by dividing it by two, then dividing the quotient by two, etc., and collecting the remainders. This method is a sort of classification scheme - odd or even.

Generalizations of the tree packing conjecture

Dániel Gerbner, Balázs Keszegh, Cory Palmer (2012)

Discussiones Mathematicae Graph Theory

The Gyárfás tree packing conjecture asserts that any set of trees with 2,3,...,k vertices has an (edge-disjoint) packing into the complete graph on k vertices. Gyárfás and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into k-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any k-chromatic graph. We also consider several other generalizations of the conjecture.

Generalized connectivity of some total graphs

Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)

Czechoslovak Mathematical Journal

We study the generalized k -connectivity κ k ( G ) as introduced by Hager in 1985, as well as the more recently introduced generalized k -edge-connectivity λ k ( G ) . We determine the exact value of κ k ( G ) and λ k ( G ) for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case k = 3 .

Graphic sequences of trees and a problem of Frobenius

Gautam Gupta, Puneet Joshi, Amitabha Tripathi (2007)

Czechoslovak Mathematical Journal

We give a necessary and sufficient condition for the existence of a tree of order n with a given degree set. We relate this to a well-known linear Diophantine problem of Frobenius.

Graphs with 3-Rainbow Index n − 1 and n − 2

Xueliang Li, Ingo Schiermeyer, Kang Yang, Yan Zhao (2015)

Discussiones Mathematicae Graph Theory

Let G = (V (G),E(G)) be a nontrivial connected graph of order n 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 set S ⊆ V (G), a tree connecting 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...

Graphs with 4-Rainbow Index 3 and n − 1

Xueliang Li, Ingo Schiermeyer, 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 called a rainbow tree if no two edges of T receive the same color. For a vertex set 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 every set S of k vertices of V (G) is called the k-rainbow index of G, denoted by rxk(G)....

Graphs with equal domination and 2-distance domination numbers

Joanna Raczek (2011)

Discussiones Mathematicae Graph Theory

Let G = (V,E) be a graph. The distance between two vertices u and v in a connected graph G is the length of the shortest (u-v) path in G. A set D ⊆ V(G) is a dominating set if every vertex of G is at distance at most 1 from an element of D. The domination number of G is the minimum cardinality of a dominating set of G. A set D ⊆ V(G) is a 2-distance dominating set if every vertex of G is at distance at most 2 from an element of D. The 2-distance domination number of G is the minimum cardinality...

Graphs with Large Generalized (Edge-)Connectivity

Xueliang Li, Yaping Mao (2016)

Discussiones Mathematicae Graph Theory

The generalized k-connectivity κk(G) of a graph G, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized k-edge-connectivity λk(G). In this paper, graphs of order n such that [...] for even k are characterized.

Groups in which the prime graph is a tree

Maria Silvia Lucido (2002)

Bollettino dell'Unione Matematica Italiana

The prime graph Γ ⁢ G of a finite group G is defined as follows: the set of vertices is π ⁢ G , the set of primes dividing the order of G , and two vertices p , q are joined by an edge (we write p ∼ q ) if and only if there exists an element in G of order p ⁢ q . We study the groups G such that the prime graph Γ ⁢ G is a tree, proving that, in this case, π ⁢ G ≤ 8 .

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