Displaying similar documents to “Generalized colorings and avoidable orientations”

Pₘ-saturated bipartite graphs with minimum size

Aneta Dudek, A. Paweł Wojda (2004)

Discussiones Mathematicae Graph Theory

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A graph G is said to be H-saturated if G is H-free i.e., (G has no subgraph isomorphic to H) and adding any new edge to G creates a copy of H in G. In 1986 L. Kászonyi and Zs. Tuza considered the following problem: for given m and n find the minimum size sat(n;Pₘ) of Pₘ-saturated graph of order n. They gave the number sat(n;Pₘ) for n big enough. We deal with similar problem for bipartite graphs.

Equitable coloring of Kneser graphs

Robert Fidytek, Hanna Furmańczyk, Paweł Żyliński (2009)

Discussiones Mathematicae Graph Theory

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The Kneser graph K(n,k) is the graph whose vertices correspond to k-element subsets of set {1,2,...,n} and two vertices are adjacent if and only if they represent disjoint subsets. In this paper we study the problem of equitable coloring of Kneser graphs, namely, we establish the equitable chromatic number for graphs K(n,2) and K(n,3). In addition, for sufficiently large n, a tight upper bound on equitable chromatic number of graph K(n,k) is given. Finally, the cases of K(2k,k) and K(2k+1,k)...

Generalized circular colouring of graphs

Peter Mihók, Janka Oravcová, Roman Soták (2011)

Discussiones Mathematicae Graph Theory

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Let P be a graph property and r,s ∈ N, r ≥ s. A strong circular (P,r,s)-colouring of a graph G is an assignment f:V(G) → {0,1,...,r-1}, such that the edges uv ∈ E(G) satisfying |f(u)-f(v)| < s or |f(u)-f(v)| > r - s, induce a subgraph of G with the propery P. In this paper we present some basic results on strong circular (P,r,s)-colourings. We introduce the strong circular P-chromatic number of a graph and we determine the strong circular P-chromatic number of complete graphs for...

Every 8-Traceable Oriented Graph Is Traceable

Susan A. van Aardt (2017)

Discussiones Mathematicae Graph Theory

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A digraph of order n is k-traceable if n ≥ k and each of its induced subdigraphs of order k is traceable. It is known that if 2 ≤ k ≤ 6, every k-traceable oriented graph is traceable but for k = 7 and for each k ≥ 9, there exist k-traceable oriented graphs that are nontraceable. We show that every 8-traceable oriented graph is traceable.

α-Labelings of a Class of Generalized Petersen Graphs

Anna Benini, Anita Pasotti (2015)

Discussiones Mathematicae Graph Theory

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An α-labeling of a bipartite graph Γ of size e is an injective function f : V (Γ) → {0, 1, 2, . . . , e} such that {|ƒ(x) − ƒ(y)| : [x, y] ∈ E(Γ)} = {1, 2, . . . , e} and with the property that its maximum value on one of the two bipartite sets does not reach its minimum on the other one. We prove that the generalized Petersen graph PSn,3 admits an α-labeling for any integer n ≥ 1 confirming that the conjecture posed by Vietri in [10] is true. In such a way we obtain an infinite class...

Some applications of pq-groups in graph theory

Geoffrey Exoo (2004)

Discussiones Mathematicae Graph Theory

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We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers.

The list linear arboricity of planar graphs

Xinhui An, Baoyindureng Wu (2009)

Discussiones Mathematicae Graph Theory

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The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having Δ ≥ 13, or for any planar graph with Δ ≥ 7 and without i-cycles for some i ∈ {3,4,5}. We also prove that ⌈½Δ(G)⌉ ≤ lla(G) ≤ ⌈½(Δ(G)+1)⌉ for any planar graph having Δ ≥ 9. ...

The edge C₄ graph of some graph classes

Manju K. Menon, A. Vijayakumar (2010)

Discussiones Mathematicae Graph Theory

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The edge C₄ graph of a graph G, E₄(G) is a graph whose vertices are the edges of G and two vertices in E₄(G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C₄. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C₄ graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E₄(G). It is shown that for any graph G without isolated vertices,...

Simplicial and nonsimplicial complete subgraphs

Terry A. McKee (2011)

Discussiones Mathematicae Graph Theory

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Define a complete subgraph Q to be simplicial in a graph G when Q is contained in exactly one maximal complete subgraph ('maxclique') of G; otherwise, Q is nonsimplicial. Several graph classes-including strong p-Helly graphs and strongly chordal graphs-are shown to have pairs of peculiarly related new characterizations: (i) for every k ≤ 2, a certain property holds for the complete subgraphs that are in k or more maxcliques of G, and (ii) in every induced subgraph H of G, that...