Displaying similar documents to “The crossing number of the generalized Petersen graph P [ 3 k , k ] ”

Median of a graph with respect to edges

A.P. Santhakumaran (2012)

Discussiones Mathematicae Graph Theory

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For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is d ( v ) = ∑ u ∈ V d ( v , u ) , the vertex-to-edge distance sum d₁(v) of v is d ₁ ( v ) = ∑ e ∈ E d ( v , e ) , the edge-to-vertex distance sum d₂(e) of e is d ₂ ( e ) = ∑ v ∈ V d ( e , v ) and the edge-to-edge distance sum d₃(e) of e is d ₃ ( e ) = ∑ f ∈ E d ( e , f ) . The set M(G) of all vertices v for which d(v) is minimum is the median of G; the set M₁(G) of all vertices v for which d₁(v) is minimum is the vertex-to-edge median of G; the set M₂(G) of all edges e for which d₂(e) is minimum is the edge-to-vertex...

New edge neighborhood graphs

Ali A. Ali, Salar Y. Alsardary (1997)

Czechoslovak Mathematical Journal

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Let G be an undirected simple connected graph, and e = u v be an edge of G . Let N G ( e ) be the subgraph of G induced by the set of all vertices of G which are not incident to e but are adjacent to u or v . Let 𝒩 e be the class of all graphs H such that, for some graph G , N G ( e ) ≅ H for every edge e of G . Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in 𝒩 e . Balasubramanian and Alsardary [1] obtained some other graphs in 𝒩 e . In this paper we given some new graphs in 𝒩 e .

Point-distinguishing chromatic index of the union of paths

Xiang'en Chen (2014)

Czechoslovak Mathematical Journal

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Let G be a simple graph. For a general edge coloring of a graph G (i.e., not necessarily a proper edge coloring) and a vertex v of G , denote by S ( v ) the set (not a multiset) of colors used to color the edges incident to v . For a general edge coloring f of a graph G , if S ( u ) ≠ S ( v ) for any two different vertices u and v of G , then we say that f is a point-distinguishing general edge coloring of G . The minimum number of colors required for a point-distinguishing general edge coloring of G , denoted...

On k-intersection edge colourings

Rahul Muthu, N. Narayanan, C.R. Subramanian (2009)

Discussiones Mathematicae Graph Theory

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We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χ’ₖ(G). Let fₖ be defined by f ₖ ( Δ ) = m a x G : Δ ( G ) = Δ χ ' ₖ ( G ) . We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.

Labeling the vertex amalgamation of graphs

Ramon M. Figueroa-Centeno, Rikio Ichishima, Francesc A. Muntaner-Batle (2003)

Discussiones Mathematicae Graph Theory

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A graph G of size q is graceful if there exists an injective function f:V(G)→ 0,1,...,q such that each edge uv of G is labeled |f(u)-f(v)| and the resulting edge labels are distinct. Also, a (p,q) graph G with q ≥ p is harmonious if there exists an injective function f : V ( G ) → Z q such that each edge uv of G is labeled f(u) + f(v) mod q and the resulting edge labels are distinct, whereas G is felicitous if there exists an injective function f : V ( G ) → Z q + 1 such that each edge uv of G is labeled f(u) + f(v) mod...