Destroying symmetry by orienting edges: complete graphs and complete bigraphs

Frank Harary; Michael S. Jacobson

Displaying similar documents to “Destroying symmetry by orienting edges: complete graphs and complete bigraphs”

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

Symmetries of embedded complete bipartite graphs

Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn (2014)

Fundamenta Mathematicae

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We characterize which automorphisms of an arbitrary complete bipartite graph $K_{n, m}$ can be induced by a homeomorphism of some embedding of the graph in S³.

On 2-periodic graphs of a certain graph operator

Ivan Havel, Bohdan Zelinka (2001)

Discussiones Mathematicae Graph Theory

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We deal with the graph operator $\bar{P o w ₂}$ defined to be the complement of the square of a graph: $\bar{P o w ₂} (G) = \bar{P o w ₂ (G)}$ . Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m, n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective...

On Ramsey $(K_{1, 2}, K ₙ)$ -minimal graphs

Mariusz Hałuszczak (2012)

Discussiones Mathematicae Graph Theory

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Let F be a graph and let , denote nonempty families of graphs. We write F → (,) if in any 2-coloring of edges of F with red and blue, there is a red subgraph isomorphic to some graph from G or a blue subgraph isomorphic to some graph from H. The graph F without isolated vertices is said to be a (,)-minimal graph if F → (,) and F - e not → (,) for every e ∈ E(F). We present a technique which allows to generate infinite family of (,)-minimal graphs if we know some special graphs. In particular,...

Clopen graphs

Stefan Geschke (2013)

Fundamenta Mathematicae

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A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their...

Difference labelling of cacti

Martin Sonntag (2003)

Discussiones Mathematicae Graph Theory

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A graph G is a difference graph iff there exists S ⊂ IN⁺ such that G is isomorphic to the graph DG(S) = (V,E), where V = S and E = i,j:i,j ∈ V ∧ |i-j| ∈ V. It is known that trees, cycles, complete graphs, the complete bipartite graphs $K_{n, n}$ and $K_{n, n - 1}$ , pyramids and n-sided prisms (n ≥ 4) are difference graphs (cf. [4]). Giving a special labelling algorithm, we prove that cacti with a girth of at least 6 are difference graphs, too.

Uniquely partitionable graphs

Jozef Bucko, Marietjie Frick, Peter Mihók, Roman Vasky (1997)

Discussiones Mathematicae Graph Theory

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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph $G [V_{i}]$ induced by $V_{i}$ has property $_{i}$ ; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property...

A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

Yury Metelsky, Kseniya Schemeleva, Frank Werner (2017)

Discussiones Mathematicae Graph Theory

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We characterize the class [...] L32 $L_{3}^{2}$ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 $L_{3}^{2}$ in the class of threshold graphs, where n is the number of vertices of a tested graph.

On hereditary properties of composition graphs

Vadim E. Levit, Eugen Mandrescu (1998)

Discussiones Mathematicae Graph Theory

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The composition graph of a family of n+1 disjoint graphs $H_{i} : 0 \leq i \leq n$ 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 $H_{i} : 0 \leq i \leq n$ 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),...

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 $𝒩_{e}$ .

Edge-connectivity of strong products of graphs

Bostjan Bresar, Simon Spacapan (2007)

Discussiones Mathematicae Graph Theory

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_{i} = y_{i}$ or $x_{i} y_{i} \in E (G_{i})$ . In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

On integral sum graphs with a saturated vertex

Zhibo Chen (2010)

Czechoslovak Mathematical Journal

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As introduced by F. Harary in 1994, a graph $G$ is said to be an $i n t e g r a l$ $s u m$ $g r a p h$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$ , $u v$ is an edge of $G$ if and only if $f (u) + f (v) = f (w)$ for some vertex $w$ in $G$ . We prove that every integral sum graph with a saturated vertex, except the complete graph $K_{3}$ , has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be if it is adjacent to every...

$T$ -preserving homomorphisms of oriented graphs

Jaroslav Nešetřil, Eric Sopena, Laurence Vignal (1997)

Commentationes Mathematicae Universitatis Carolinae

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A homomorphism of an oriented graph $G = (V, A)$ to an oriented graph $G^{'} = (V^{'}, A^{'})$ is a mapping $ϕ$ from $V$ to $V^{'}$ such that $ϕ (u) ϕ (v)$ is an arc in $G^{'}$ whenever $u v$ is an arc in $G$ . A homomorphism of $G$ to $G^{'}$ is said to be $T$ -preserving for some oriented graph $T$ if for every connected subgraph $H$ of $G$ isomorphic to a subgraph of $T$ , $H$ is isomorphic to its homomorphic image in $G^{'}$ . The $T$ -preserving oriented chromatic number ${\vec{χ}}_{T} (G)$ of an oriented graph $G$ is the minimum number of vertices in an oriented graph $G^{'}$ such that there exists a $T$ -preserving...

ℓ²-homology and planar graphs

Timothy A. Schroeder (2013)

Colloquium Mathematicae

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In his 1930 paper, Kuratowski proves that a finite graph Γ is planar if and only if it does not contain a subgraph that is homeomorphic to K₅, the complete graph on five vertices, or $K_{3, 3}$ , the complete bipartite graph on six vertices. This result is also attributed to Pontryagin. In this paper we present an ℓ²-homological method for detecting non-planar graphs. More specifically, we view a graph Γ as the nerve of a related Coxeter system and construct the associated Davis complex, $Σ_{Γ}$ . We...