Results 31 to 40 of about 3,613,364 (137)
Saturated Partial Embeddings of Planar Graphs
Journal of Graph Theory, EarlyView.ABSTRACT In this work, we study how far one can deviate from optimal behavior when embedding a planar graph. For a planar graph G $G$, we say that a plane subgraph H ⊆ G $H\subseteq G$ is a plane‐saturated subgraph if adding any edge (possibly with new vertices) to H $H$ would either violate planarity or make the resulting graph no longer a subgraph of
Alexander Clifton, Nika Salia
wiley +1 more source
Equivalent Formulation of Thomassen's Conjecture Using Tutte Paths in Claw‐Free Graphs
Journal of Graph Theory, EarlyView.ABSTRACT We continue studying Thomassen's conjecture (every 4‐connected line graph has a Hamilton cycle) in the direction of a recently shown equivalence with Jackson's conjecture (every 2‐connected claw‐free graph has a Tutte cycle), and we extend the equivalent formulation as follows: In every connected claw‐free graph, any two vertices are connected
Adam Kabela +2 more
wiley +1 more source
A Note on Lovász Characterization of Perfect Graphs
Journal of Graph Theory, EarlyView.ABSTRACT A graph is perfect if, for every induced subgraph, the chromatic number equals the size of its largest clique. In 1972, Lovász established a fundamental characterization of perfect graphs, showing that a graph is perfect if and only if, for every induced subgraph, the product of the size of the largest independent set and the size of the ...
James Alex
wiley +1 more source
Characterization of Graphs Without Even F $F$‐Orientations
Journal of Graph Theory, EarlyView.ABSTRACT A graph G $G$ is 1‐extendable if every edge belongs to at least one 1‐factor of G $G$. Let G $G$ be a graph with a 1‐factor F $F$. Then an even (odd) F $F$ ‐orientation of G $G$ is an orientation in which each F $F$‐alternating cycle has exactly an even (odd) number of edges directed in the same fixed direction around the cycle.
Marién Abreu +3 more
wiley +1 more source
Line game-perfect graphs [PDF]
Discrete Mathematics & Theoretical Computer ScienceThe $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$.
Stephan Dominique Andres, Wai Lam Fong
doaj +1 more source
Journal of Graph Theory, EarlyView.ABSTRACT A 2‐edge‐coloured graph G $G$ is called locally complete if for each vertex v $v$, the vertices adjacent to v $v$ through edges of the same colour induce a complete subgraph in G $G$. Locally complete 2‐edge‐coloured graphs have nice properties, and there exists a polynomial algorithm to decide whether such a graph has an alternating ...
Jørgen Bang‐Jensen, Jing Huang
wiley +1 more source
Vertex colouring and forbidden subgraphs - a survey [PDF]
, 2004 There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph ...
Randerath, Bert, Schiermeyer, Ingo
core
The 1 3 $\frac{1}{3}$‐Conjectures for Domination in Cubic Graphs
Journal of Graph Theory, EarlyView.ABSTRACT A set S $S$ of vertices in a graph G $G$ is a dominating set of G $G$ if every vertex not in S $S$ is adjacent to a vertex in S $S$. The domination number of G $G$, denoted by γ ( G ) $\gamma (G)$, is the minimum cardinality among all dominating sets in G $G$. In a breakthrough paper in 2008, Löwenstein and Rautenbach proved that if G $G$ is a
Paul Dorbec, Michael A. Henning
wiley +1 more source
The Minimal Nonplanar Strong Digraphs
Journal of Graph Theory, EarlyView.ABSTRACT Kuratowski's theorem says that the minimal (under subgraph containment) graphs that are not planar are the subdivisions of K 5 ${K}_{5}$ and of K 3 , 3 ${K}_{3,3}$. Here we study the minimal (under subdigraph containment) strongly‐connected digraphs that are not planar.
Stephen Bartell, Paul Seymour
wiley +1 more source
A Structural Description of Zykov and Blanche Descartes Graphs
Journal of Graph Theory, EarlyView.ABSTRACT Around 1950, Zykov and Blanche Descartes proposed the two first explicit constructions of triangle‐free graphs with arbitrarily large chromatic number. We define a Zykov (resp. Blanche Descartes) graph as any induced subgraph of a graph created using Zykov's construction (resp. Blanche Descartes's).
Malory Marin +3 more
wiley +1 more source