Results 51 to 60 of about 380,403 (285)
Tuza's Conjecture for Threshold Graphs [PDF]
Discrete Mathematics & Theoretical Computer Science, 2022 Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including
Marthe Bonamy +6 more
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Straight-line Drawings of 1-Planar Graphs [PDF]
arXiv, 2021 A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. However, there are 1-planar graphs which do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges, so that edges of the same color do not cross.
arxiv
A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable
Discrete Dynamics in Nature and Society, 2020 A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color.
Enqiang Zhu, Yongsheng Rao
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Planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable [PDF]
Discrete Mathematics & Theoretical Computer Science, 2016 For planar graphs, we consider the problems of list edge coloring and list total coloring. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors.
Marthe Bonamy +2 more
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On Edge Colorings of 1-Planar Graphs without 5-Cycles with Two Chords
Discussiones Mathematicae Graph Theory, 2019 A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-planar graph with maximum degree ∆ ≥ 8 is edge-colorable with ∆ colors if each of its 5-cycles contains ...
Sun Lin, Wu Jianliang
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Longer Cycles in Essentially 4-Connected Planar Graphs
Discussiones Mathematicae Graph Theory, 2020 A planar 3-connected graph G is called essentially 4-connected if, for every 3-separator S, at least one of the two components of G − S is an isolated vertex.
Fabrici Igor +3 more
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The maximum matching extendability and factor-criticality of 1-planar graphs [PDF]
arXiv, 2022 A graph is $1$-$planar$ if it can be drawn in the plane so that each edge is crossed by at most one other edge. Moreover, a 1-planar graph $G$ is $optimal$ if it satisfies $|E(G)|=4|V(G)|-8$. J. Fujisawa et al. [16] first considered matching extension of optimal 1-planar graphs, obtained that each optimal 1-planar graph of even order is 1-extendable ...
arxiv
Equitable Coloring and Equitable Choosability of Planar Graphs without chordal 4- and 6-Cycles [PDF]
Discrete Mathematics & Theoretical Computer Science, 2019 A graph $G$ is equitably $k$-choosable if, for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices.
Aijun Dong, Jianliang Wu
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A structure of 1-planar graph and its applications to coloring problems [PDF]
Graphs and Combinatorics, 35(3) (2019) 677-688, 2019 A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the $(p,1)$-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the
arxiv +1 more source
Planar Graphs of Maximum Degree 6 and without Adjacent 8-Cycles Are 6-Edge-Colorable
Journal of Mathematics, 2021 In this paper, by applying the discharging method, we show that if G is a planar graph with a maximum degree of Δ=6 that does not contain any adjacent 8-cycles, then G is of class 1.
Wenwen Zhang
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