1-planar graphs with girth at least 6 are (1,1,1,1)-colorable

Lili Song; Lei Sun

Displaying similar documents to “1-planar graphs with girth at least 6 are (1,1,1,1)-colorable”

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...

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...

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.

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 $𝒩_{e}$ .

Graceful signed graphs

Mukti Acharya, Tarkeshwar Singh (2004)

Czechoslovak Mathematical Journal

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A $(p, q)$ -sigraph $S$ is an ordered pair $(G, s)$ where $G = (V, E)$ is a $(p, q)$ -graph and $s$ is a function which assigns to each edge of $G$ a positive or a negative sign. Let the sets $E^{+}$ and $E^{-}$ consist of $m$ positive and $n$ negative edges of $G$ , respectively, where $m + n = q$ . Given positive integers $k$ and $d$ , $S$ is said to be $(k, d)$ -graceful if the vertices of $G$ can be labeled with distinct integers from the set ${0, 1, \dots, k + (q - 1) d}$ such that when each edge $u v$ of $G$ is assigned the product of its sign and the absolute difference of the integers assigned to...

Acyclic reducible bounds for outerplanar graphs

Mieczysław Borowiecki, Anna Fiedorowicz, Mariusz Hałuszczak (2009)

Discussiones Mathematicae Graph Theory

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For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. $G [V_{i}] \in_{i}$ for i = 1,...,n, 2. for every pair i,j of distinct colours the subgraph induced in G by the set of edges uv such that $u \in V_{i}$ and $v \in V_{j}$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring....

On light subgraphs in plane graphs of minimum degree five

Stanislav Jendrol&#039;, Tomáš Madaras (1996)

Discussiones Mathematicae Graph Theory

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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 $K_{1, 3}$ and $K_{1, 4}$ and a light 4-path P₄. The results obtained for $K_{1, 3}$ and P₄ are best possible.

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...

4-cycle properties for characterizing rectagraphs and hypercubes

Khadra Bouanane, Abdelhafid Berrachedi (2017)

Czechoslovak Mathematical Journal

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A $(0, 2)$ -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0, λ)$ -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0, 2)$ -graph. $(0, 2)$ -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0, λ)$ -graphs and more specifically...