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Weak Degeneracy of Planar Graphs and Locally Planar Graphs [PDF]
The Electronic Journal of Combinatorics, 2023 Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d+1)$-list assignment $L$ of $G$, one can construct an $L$ coloring of $G$ by a modified ...
Ming Han +4 more
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Discrete Mathematics, 1974 AbstractA 3-valent graph G is cyclically n-connected provided one must cut at least n edges in order to separate any two circuits of G. If G is cyclically n-connected but any separation of G by cutting n edges yields a component consisting of a simple circuit, then we say that G is strongly cyclically n-connected.
David Barnette
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Historia Mathematica, 1985 AbstractIn the late 1920s several mathematicians were on the verge of discovering a theorem for characterizing planar graphs. The proof of such a theorem was published in 1930 by Kazimierz Kuratowski, and soon thereafter the theorem was referred to as the Kuratowski Theorem. It has since become the most frequently cited result in graph theory. Recently,
John W. Kennedy +2 more
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Contracting a Planar Graph Efficiently [PDF]
25th Annual European Symposium on Algorithms (ESA 2017), 2017 We present a data structure that can maintain a simple planar graph under edge contractions in linear total time. The data structure supports adjacency queries and provides access to neighbor lists in $O(1)$ time.
J. Holm +5 more
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A theorem on planar graphs [PDF]
Transactions of the American Mathematical Society, 1956 W. T. Tutte
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The Odd Chromatic Number of a Planar Graph is at Most 8 [PDF]
Graphs and Combinatorics, 2022 Petruševski and Škrekovski recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph G is said to be odd if for each non-isolated vertex $$x \in V(G)$$ x ∈ V ( G ) there exists a colour c appearing an odd number ...
J. Petr, Julien Portier
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The structure and the list 3-dynamic coloring of outer-1-planar graphs [PDF]
Discrete Mathematics & Theoretical Computer Science, 2021 An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge.
Yan Li, Xin Zhang
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Kramers–Wannier Duality and Random-Bond Ising Model [PDF]
EntropyWe present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators ...
Chaoming Song
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An improved planar graph product structure theorem [PDF]
Electronic Journal of Combinatorics, 2021 Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every planar graph $G$ there is a graph $H$ with treewidth at most 8 and a path $P$ such that $G\subseteq H\boxtimes P$. We improve this result by replacing "treewidth at most
T. Ueckerdt, D. Wood, Wendy Yi
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Improved product structure for graphs on surfaces [PDF]
Discrete Mathematics & Theoretical Computer Science, 2022 Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve
Marc Distel +3 more
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