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In his PhD thesis [Structural aspects of switching classes, Leiden Institute of Advanced Computer Science, 2001] Hage posed the following problem: “characterize the maximum size graphs in switching classes”. These are called s-maximal graphs. In this paper, we study the properties of such graphs. In particular, we show that any graph with sufficiently large minimum degree is s-maximal, we prove that join of two s-maximal graphs is also an s-maximal graph, we give complete characterization of triangle-free...
The composition graph of a family of n+1 disjoint graphs is the graph H obtained by substituting the n vertices of H₀ respectively by the graphs H₁,H₂,...,Hₙ. If H has some hereditary property P, then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors have a certain common P to endow H with this P. For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978), and the...
In this Note, we study infinite graphs with locally finite outerplane embeddings, given a characterization by forbidden subgraphs.
A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property is of finite character if a graph G has a property if and only if every finite induced subgraph of G has a property . Let ₁,₂,...,ₙ be graph properties of finite character, a graph G is said to be (uniquely) (₁, ₂, ...,ₙ)-partitionable if there is an (exactly one) partition V₁, V₂, ..., Vₙ of V(G) such that for i = 1,2,...,n. Let us denote by ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ the class of all (₁,₂,...,ₙ)-partitionable...
The concept of the -pairable graphs was introduced by Zhibo Chen (On -pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter , called the pair length of a graph , as the maximum such that is -pairable and if is not -pairable for any positive integer . In this paper, we answer the two open questions raised by Chen in the case that the graphs involved...
A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is well known that a plane graph of minimum degree five contains light edges and light triangles. In this paper we show that every plane graph of minimum degree five contains also light stars and and a light 4-path P₄. The results obtained for and P₄ are best possible.
A signed graph (or sigraph for short) is an ordered pair S = (Su,σ), where Su is a graph, G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a sigraph S its •-line sigraph, L•(S) is the sigraph in which the edges of S are represented as vertices, two of these vertices are defined adjacent whenever the corresponding edges in S have a vertex in common, any such L-edge ee′ has the sign given by the product of the signs of the...
A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. We characterize magic line graphs of general graphs and describe some class of supermagic line graphs of bipartite graphs.
A graph G is a (Kq, k) stable graph (q ≥ 3) if it contains a Kq after deleting any subset of k vertices (k ≥ 0). Andrzej ˙ Zak in the paper On (Kq; k)-stable graphs, ( doi:/10.1002/jgt.21705) has proved a conjecture of Dudek, Szyma´nski and Zwonek stating that for sufficiently large k the number of edges of a minimum (Kq, k) stable graph is (2q − 3)(k + 1) and that such a graph is isomorphic to sK2q−2 + tK2q−3 where s and t are integers such that s(q − 1) + t(q − 2) − 1 = k. We have proved (Fouquet...
A graph is called a partial cube if it admits an isometric embedding into a hypercube. Subdivisions of wheels are considered with respect to such embeddings and with respect to the convexity of their intervals. This allows us to answer in negative a question of Chepoi and Tardif from 1994 whether all bipartite graphs with convex intervals are partial cubes. On a positive side we prove that a graph which is bipartite, has convex intervals, and is not a partial cube, always contains a subdivision...
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