Displaying similar documents to “Light paths with an odd number of vertices in polyhedral maps”

Paths with restricted degrees of their vertices in planar graphs

Stanislav Jendroľ (1999)

Czechoslovak Mathematical Journal

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In this paper it is proved that every 3 -connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23 . Analogous results are stated for 3 -connected planar graphs of minimum degree 4 and 5 . Moreover, for every pair of integers n 3 , k 4 there is a 2 -connected planar graph such that every path on n vertices in it has a vertex of degree k .

Classifying trees with edge-deleted central appendage number 2

Shubhangi Stalder, Linda Eroh, John Koker, Hosien S. Moghadam, Steven J. Winters (2009)

Mathematica Bohemica

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The eccentricity of a vertex v of a connected graph G is the distance from v to a vertex farthest from v in G . The center of G is the subgraph of G induced by the vertices having minimum eccentricity. For a vertex v in a 2-edge-connected graph G , the edge-deleted eccentricity of v is defined to be the maximum eccentricity of v in G - e over all edges e of G . The edge-deleted center of G is the subgraph induced by those vertices of G having minimum edge-deleted eccentricity. The edge-deleted...

Light classes of generalized stars in polyhedral maps on surfaces

Stanislav Jendrol', Heinz-Jürgen Voss (2004)

Discussiones Mathematicae Graph Theory

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A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. S i denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(), on d + 2- m()...

Vertex-disjoint stars in graphs

Katsuhiro Ota (2001)

Discussiones Mathematicae Graph Theory

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In this paper, we give a sufficient condition for a graph to contain vertex-disjoint stars of a given size. It is proved that if the minimum degree of the graph is at least k+t-1 and the order is at least (t+1)k + O(t²), then the graph contains k vertex-disjoint copies of a star K 1 , t . The condition on the minimum degree is sharp, and there is an example showing that the term O(t²) for the number of uncovered vertices is necessary in a sense.

Bounds concerning the alliance number

Grady Bullington, Linda Eroh, Steven J. Winters (2009)

Mathematica Bohemica

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P. Kristiansen, S. M. Hedetniemi, and S. T. Hedetniemi, in Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004), 157–177, and T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, in Global defensive alliances in graphs, Electron. J. Combin. 10 (2003), introduced the defensive alliance number a ( G ) , strong defensive alliance number a ^ ( G ) , and global defensive alliance number γ a ( G ) . In this paper, we consider relationships between these parameters and the domination number γ ( G ) . For any positive...

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 ( v ) = e E d ( v , e ) , the edge-to-vertex distance sum d₂(e) of e is d ( e ) = v V d ( e , v ) and the edge-to-edge distance sum d₃(e) of e is d ( 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...

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...

The eavesdropping number of a graph

Jeffrey L. Stuart (2009)

Czechoslovak Mathematical Journal

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Let G be a connected, undirected graph without loops and without multiple edges. For a pair of distinct vertices u and v , a minimum { u , v } -separating set is a smallest set of edges in G whose removal disconnects u and v . The edge connectivity of G , denoted λ ( G ) , is defined to be the minimum cardinality of a minimum { u , v } -separating set as u and v range over all pairs of distinct vertices in G . We introduce and investigate the eavesdropping number, denoted ε ( G ) , which is defined to be the maximum cardinality...