Zsolt Patakfalvi
A graph is called normal if its vertex set can be covered by cliques and also by stable sets, such that every such clique and stable set have non-empty intersection. This notion is due to Korner, who introduced the class of normal graphs as an extension of the class of perfect graphs. Normality has also relevance in information theory. Here we prove, that the line graphs of cubic graphs are normal.
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Normal 6-edge-colorings of cubic graphs with oddness 2
本文推广了Mazzuoccolo和Mkrtchyan关于循环置换图正常6-边着色的结果，证明了每个奇度为2的三次图都允许用至多6种颜色进行正常边着色。
In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter $μ_3$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $|E(G)|-μ_3(G)$, …
We present results on the relative realizability of infinite families of lines on general smooth tropical cubic surfaces. Inspired by the problem of relative realizability of lines on surfaces, we investigate the information we can derive tropically from the Brundu-Logar normal form of smooth cubic surfaces. In particular, we prove that for a residue field of characteristic $\neq 2$ the tropicalization of the Brundu-Logar normal form is not smooth. We also take first steps in investigating the behavior of the tropicalized lines.
In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors such that each edge of the graph is normal. We denote by $χ'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conje…
A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors having the additional property that when looking at the set of colors assigned to any edge $e$ and the four edges adjacent it, we have either exactly five distinct colors or exactly three distinct colors. We denote by $χ'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal $k$-edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. More precisely, it is known that proving $χ'_{N}(G)\leq 5$ for every bridgeless cubic graph is equivalent to…
List strong and list normal edge-coloring of (sub)cubic graphs
研究了立方图的强边着色和正常边着色的列表版本，证明了最大度数为3的图的列表强色数不超过10，并展示了存在具有不同强色数和其列表版本的图。
The groups which can act semisymmetrically on a cubic graph of twice odd order are determined modulo a normal subgroup which acts semiregularly on the vertices of the graph.
Normal 5-edge coloring of some more snarks superpositioned by the Petersen graph
本文延续此前关于Petersen图叠加snark图正规5边着色的研究，验证了剩余所有超顶点与超边连接方式下Petersen着色猜想均成立，推进了该猜想的证明范围。
We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.
We provide precise asymptotic estimates for the number of several classes of labelled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky et al. (Random Structures Algorithms 2007). We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptoticall…
We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.
We consider the structures of the plane-line and point-line incidence matrices of the projective space $\mathrm{PG}(3,q)$ connected with orbits of planes, points, and lines under the stabilizer group of the twisted cubic. In the literature, lines are partitioned into classes, each of which is a union of line orbits. In this paper, for all $q$, even and odd, we determine the incidence matrices connected with a family of orbits of the class named $\mathcal{O}_6$. This class contains lines external to the twisted cubic. The considered family include an essential part of all $\mathcal{O}_6$ orbits…
Decomposing cubic graphs into isomorphic linear forests
针对Wormald 1987年提出的“4n顶点立方图边可划分为两个同构线性森林”的猜想，本文利用概率工具、结构分析及局部重着色技术，证明其对大型连通立方图成立。
In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of $\mathrm{PG}(3,q)$ are partitioned into classes, each of which is a union of line orbits. In this paper, all classes of lines consisting of a unique orbit are found. For the remaining line types, with one exception, it is proved that they consist exactly of two or three orbits; sizes and structures of these orbits are determined. Also, the subgroups of the stabilizer group of the twisted cubic fixing lines of the orbits are obtained. Problems …
Orbits and incidence matrices for points, planes and lines regarding the twisted cubic in PG(3,q), q = 2, 3, 4
本文研究了有限域F_q上三维射影空间PG(3,q)中扭曲三次曲线的轨道分类及对应的incidence矩阵，针对q=2,3,4的情况解决开放问题。
The cuspidal cubic and line arrangements with only triple points
本文构造了射影平面中仅含三重点奇点的新无限直线排列族，并重新得到已知例子。
On cubic-line arrangements with simple singularities
研究复射影平面中含节点、普通三重点和 $A_{5}$ 奇点的三次曲线-直线排列的组合与代数性质，给出 Hirzebruch 型不等式并几乎完成自由性分类。
In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by 3. These equivalence classes are considered as orbits of lines in $\mathrm{PG}(3,q)$, under the action of the subgroup $G\cong \mathrm{PGL}(2,q)$ of $\mathrm{PGL}(4,q)$ which preserves the twisted cubic $\mathcal{C}$ in $\mathrm{PG}(3,q)$. In particular we determine the point orbit distributions and plane orbit distributions of all $G$-orbits of lines which are contained in an osculating plane of $\mathcal{C}$, have non-empty intersection w…
A Near-Linear-Time Algorithm for Finding a Well-Spread Perfect Matching in Bridgeless Cubic Graphs
本文提出一种近线性时间算法，用于在无桥立方图中找到完美匹配，该匹配与每个3边割恰好相交一次，改进了Boyd等人和之前算法的效率。
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