Rishi Advani
There is empirical evidence supporting the claim that almost all cubic non-Hamiltonian graphs are bridge graphs. In this paper, we pose a related conjecture and prove that the original claim holds for non-3-connected graphs if the conjecture is true.
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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 well-known conjecture by Lovász and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $cγ^n$,…
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。
It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an $n$-vertex planar graph, augments this graph in $O(n^2)$ steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized Antifactor-problem for the fixed embedding case, and SPQR-trees for the variab…
A superlinear bound on the number of perfect matchings in cubic bridgeless graphs
Lovasz和Plummer在20世纪70年代提出，立方无桥图有指数级的完美匹配。Voorhoeve在1979年对二部图验证了这个猜想，Chudnovsky和Seymour在2008年对可平面图验证了这个猜想，但一般来说只有线性边界是已知的。本文给出了一般情况下的第一超线性界。
Covering Bridgeless Claw-Free Cubic Graphs with Perfect Matchings
本文证明Berge猜想对无桥无爪三次图成立，并给出四个完美匹配覆盖至少299/315边数的结果。
On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings
确定覆盖立方无桥图的边集所需的完美匹配数的问题与Berge和Fulkerson的一个著名猜想密切相关。本文证明了对一个给定的立方无桥图判定这个数是否至多为4是NP-完全的。我们还构造了一个无限族$\cal F$的snark（围长至少为5且色指数为4的循环4-边连通三次图），其边集不能被4个完美匹配覆盖.已知的此类图表只有两个。事实证明，家庭$\cal F$也有有趣的性质，最短的周期覆盖问题。任何具有$m$边的三次无桥图的最短圈覆盖的长度至少为$\tfrac43m$，并且证明了这个不等式对$\cal F$图是严格的.我们还构造了第一个已知的snark，没有长度小于$\tfrac43m+2$的循环覆盖。
A tris of perfect matchings in bridgeless claw-free cubic graphs
本文证明了Fan--Raspaud猜想对无桥无爪三次图成立，即存在三个完美匹配交集为空，并提供了与5-圈双覆盖猜想相关的信息。
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…
A conjecture of Berge and Fulkerson (1971) states that every cubic bridgeless graph contains 6 perfect matchings covering each edge precisely twice, which easily implies that every cubic bridgeless graph has three perfect matchings with empty intersection (this weaker statement was conjectured by Fan and Raspaud in 1994). Let $m_t$ be the supremum of all reals $α\le 1$ such that for every cubic bridgeless graph $G$, there exist $t$ perfect matchings of $G$ covering a fraction of at least $α$ of the edges of $G$. It is known that the Berge-Fulkerson conjecture is equivalent to the statement tha…
Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs with a 2-factor consisting of two circuits. The first part of this result implies that Berge Conjecture holds for hypohamiltonian cubic graphs.
Short Cycle Covers of Cubic Graphs and Graphs with Minimum Degree Three
Alon和Tarsi的最短圈覆盖猜想证明了所有具有$m$边的无桥图的边都能被总长度不超过$7m/5=1.400m$的圈覆盖。我们证明了每一个立方无桥图的圈覆盖的总尺子至多为$34m/21\approx 1.619m$，每一个最小度为3的无桥图的圈覆盖的总尺子至多为$44m/27\approx 1.630m$ .
On the time complexity of finding a well-spread perfect matching in bridgeless cubic graphs
本文提出一种O(n log^4 n)时间算法，用于在3-边连通三次图中寻找与每个3-边割恰好交于一条边的完美匹配，并应用于奇异边受限的图嵌入计算，支持Cycle Double Cover猜想研究。
A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs.
On some perfect matching conjectures in infinite, cubic, bridgeless graphs
本文研究Berge-Fulkerson等三个关于完美匹配的著名猜想在无限无割边立方图中的有效性，类比有限情形确立其蕴含关系，并证明各猜想的有限与无限版本等价。
We show that every edge in a 2-edge-connected planar cubic graph is either contained in a 2-edge-cut or is a chord of some cycle that is contained in a 2-factor of the graph. As a consequence, we show that every edge in a cyclically 4-edge-connected planar cubic graph, except $K_2^3$ and $K_4$, is contained in a perfect matching whose removal disconnects the graph. We obtain a complete characterization of 2-edge-connected planar cubic graphs that have an edge such that every 2-factor containing the edge is a Hamiltonian cycle, and also of those that have an edge such that the complement of eve…
Cubic graphs induced by bridge trisections
本文证明每个 Tait 着色的三次图均可作为桥三剖分的1-骨架，实现不打结曲面，并推广至不可定向情形，为三平面图转换提供新方法。
A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochro…
In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture originally proposed by the first author, which states that in every simple bridgeless cubic graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a …
Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs
在1954年，Tutte证明了每个无桥图都有一个无处为零的5-流。设$ω$是无桥三次图的2-因子中奇圈的最小数目。Tutte猜想等价于它对具有$ω\geq 2$的三次图的限制.我们证明了如果一个三次图$G$没有少于$ {5/2} ω- 1$边的边割，且该边割能分离$G$的最小2-因子的两个奇圈，则$G$有一个无零5-流.这意味着，如果一个三次图$G$是循环$n$ -边连通的，并且是$n \geq {5/2} ω- 1$，则$G$有一个无处为零的5-流。
Improved Domination--Packing Bounds in Claw-Free Cubic Graphs and Unit Disk Graphs
研究无爪三次图与单位圆盘图中支配数γ(G)与打包数ρ(G)的比值，将单位圆盘图的界从32改进到16，并证明无桥无爪三次图满足γ(G) ≤ (7/4)ρ(G) + 5/6。
Covering a cubic graph with perfect matchings
设G是无桥三次图。Berge和Fulkerson的一个著名猜想可以表述如下：存在G的五个完美匹配，使得G的每条边至少包含在其中一个中。在这里，我们证明了在每个无桥三次图中，存在5个完美匹配覆盖至少等于215/231的一部分边。通过推广这一结果，我们降低了图G的边集所需的完美匹配数的最佳已知上界（用图的大小表示）。
A cycle cover of a graph is a collection of cycles such that each edge of the graph is contained in at least one of the cycles. The length of a cycle cover is the sum of all cycle lengths in the cover. We prove that every bridgeless cubic graph with $m$ edges has a cycle cover of length at most $212/135 \cdot m \ (\approx 1.570 m)$. Moreover, if the graph is cyclically $4$-edge-connected we obtain a cover of length at most $47/30 \cdot m \approx 1.567 m$.
On covering cubic graphs with three perfect matchings
研究了立方图用三个完美匹配覆盖的效率，证明了对于特定范围内的比值，存在无限多个立方图满足该比值。
If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger ($P_{10}$-conjecture) states that for any bridgeless cubic graph $G$, one has: $P_{10}\prec G$. The Sylvester coloring conjecture ($S_{10}$-conjecture) states that for any cubic graph $G$, $S_{10}\prec G$. In this paper, we introduce two new conjectures that are related to t…
Avoiding 5-circuits in a 2-factor of cubic graphs
我们证明了除Petersen图外的所有在$n$顶点上的无桥三次图$G$都有一个2-因子，且至多有$2(n-2)/15$长度为$5$的回路.一个无限族的图达到这个界限。我们还表明，$G$有一个2-因子，最多$n/5.8\overline{3}$奇电路。这改进了先前已知的$n/5.41$的界限[Lukoka，Máčajová，Mazák，Škoviera：Small snarks with large oddness，arXiv：1212.3641 [cs.DM] ]。
Non-conflicting no-where zero $Z_2\times Z_2$ flows in cubic graphs
本文定义三次图中关于完美匹配补2-因子的无冲突无处为零 $Z_2\times Z_2$ 流，证明其存在蕴含正常6-边着色，并用于证明无爪及特定2-因子三次图的着色性质，同时构造反例并关联Thomassen猜想。
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 introduce weak oddness $ω_{\textrm w}$, a new measure of uncolourability of cubic graphs, defined as the least number of odd components in an even factor. For every bridgeless cubic graph $G$, $ρ(G)\leω_{\textrm w}(G)\leω(G)$, where $ρ(G)$ denotes the resistance of $G$ and $ω(G)$ denotes the oddness of $G$, so this new measure is an approximation of both oddness and resistance. We demonstrate that there are graphs $G$ satisfying $ρ(G) < ω_{\textrm w}(G) < ω(G)$, and that the difference between any two of those three measures can be arbitrarily large. The construction implies that if we repl…
The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on cores and weak cores of a bridgeless cubic graph. In particular, we prove that the Fan-Raspaud Conjecture is equivalent to a conjecture proposed in [E. Steffen, 1-factor and cycle covers of cubic gr…
Let $G$ be a bridgeless cubic graph. The Berge--Fulkerson Conjecture (1970s) states that $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan--Raspaud Conjecture (1994), which states that $G$ admits three perfect matchings such that every edge of $G$ belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that $G$ admits two perfect matchings whose deletion yields a bipartite subgraph of $G$. It can be shown th…
An improved linear bound on the number of perfect matchings in cubic graphs
证明了任意n阶立方无桥图至少有3 n/4-10个完美匹配。这是第一个与完美匹配多面体的最大维数相差超过一个常数的界限。
Tutte's 5-Flow Conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent to its restriction on cyclically 6-edge connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero 5-flow.
Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $μ_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if $μ_3(G) \not = 0$, then $2 μ_3(G)$ is an upper bound for the girth of $G$. We also prove some new upper bounds for the length …
Counterexamples to the Cubic Graph Domination Conjecture
设v（G）和dom（G）表示图G的顶点数和控制数，r（G）= dom（G）/v（G）$. Let [x] and ]x[ be the floor and the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic graph, then dom(G) is at most ]v(G)/3[. In 2005 A. Kostochka and B. Stodolsky disproved this conjecture for cubic graphs of connectivity one and maintained that the conjecture may still be true for 2-connected cubic graphs. Their minimum counterexample C has 4 bridges, v(C) = 60, anddom (C) = 21. In this paper we disprove Reed's conjecture for 2-connected cubic graphs by providing a sequence (R(k): k > 2) of cubic graphs of connectivity two w…
Berge's conjecture for cubic graphs with small colouring defect
本文证明所有着色缺陷为 3 的无桥三次图均可由至多五个完美匹配覆盖，并对循环 4-边连通情形加强为四个（Petersen 图除外）。
A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. Lukot'ka, Mácajová, Mazák and Škoviera showed in [Electron. J. Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and remarked that there are exactly two such graphs of that order. However, this remark is incorrect as -- using an exhaustive computer search -- we show that there are in fact three snarks with oddness 4 on 28 vertices. In this note we present the missing snark and also determine all…
Directed Cycle Double Cover Conjecture: Fork Graphs
我们探讨著名的Jaeger的有向圈双覆盖猜想，这是等价于断言，每一个立方无桥图有一个嵌入在一个封闭的可定向曲面，没有对偶回路。我们将每个三次图G与一个我们称之为“六边形图”的新对象H相关联; H的完美匹配描述了G在闭合可定向曲面上的所有嵌入。对六边形图的研究使我们定义了一类新的图，我们称之为“瘦叉图”。叉图是通过顺序连接叉型图并执行Y-Delta、Delta-Y变换而从三角形获得的立方无桥图;瘦叉图是满足连通性的叉图。我们证明了Jaeger猜想对一类瘦叉图成立。瘦叉图类是丰富的，即对于每一个立方无桥图G，都有一个瘦叉图包含G的一个细分作为诱导子图。我们的结果建立第一次，尽我们所知，有效性的Jaeger猜想在一个广泛的归纳定义类的图。
Exponentially many perfect matchings in cubic graphs
我们证明了每一个立方无桥图G至少有2^（|V（G）|/3656）完美匹配。这证实了Lovasz和Plummer的一个古老猜想。这个版本的文件使用了不同的定义节从期刊版本的文件和不同的证明引理18。这简化了我们在整篇论文中的论点的阐述。
A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of $G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at most $k-2$ vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph $G$ with a circular nowhere-zero $r$-flow has a $\lfloor r \rfloor$-weak bisection. In this paper we study problems related to the existence of $k$-weak bisections. We believe that every cubic graph which ha…
Graphs, Disjoint Matchings and Some Inequalities
本文研究图的最大k边可染色子图大小ν_k(G)，推导已有不等式的蕴含关系，改进多类立方图的ν_2(G)相关下界，并给出少环图的ν_k(G)下界及相关猜想。
If $f:\mathbb{N}\rightarrow \mathbb{N}$ is a function, then let us say that $f$ is sublinear if \[\lim_{n\rightarrow +\infty}\frac{f(n)}{n}=0.\] If $G=(V,E)$ is a cubic graph and $c:E\rightarrow \{1,...,k\}$ is a proper $k$-edge-coloring of $G$, then an edge $e=uv$ of $G$ is poor (rich) in $c$, if the edges incident to $u$ and $v$ are colored with three (five) colors. An edge is abnormal if it is neither rich nor poor. The Petersen coloring conjecture of Jaeger states that any bridgeless cubic graph admits a proper 5-edge-coloring $c$, such that there is no an abnormal edge of $G$ with respect…
Bouchet conjectured in 1983 that every flow-admissible signed graph admits a nowhere-zero 6-flow which is equivalent to the restriction to cubic signed graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable cubic signed graph admits a nowhere-zero $10$-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $10$-flow. As a byproduct, we also show that every flow-admissible hamiltonian signed graph admits a nowhere-zero $8$-flow.
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 classical Berge-Fulkerson conjecture states that any bridgeless cubic graph $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to two of the perfect matchings from the list. In this short note, we discuss two statements that are consequences of this conjecture. We show that the first statement is equivalent to Fan-Raspaud conjecture. We also show that the smallest counter-example to the second one is a cyclically $4$-edge-connected cubic graph.
If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger states that for any bridgeless cubic graph $G$, one has: $P\prec G$. The second author has recently introduced the Sylvester coloring conjecture, which states that for any cubic graph $G$ one has: $S\prec G$. Here $S$ is the Sylvester graph on $10$ vertices. In this paper, w…
On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers
本文研究了3连通平面图的可定向回路双覆盖特性，证明了其与Apollonian网络对偶图的关系，并扩展了这一结论至所有3连通平面图。
Some new results on Sylvester colorings of cubic graphs
研究三次图的西尔维斯特着色，通过定义相关映射和特殊三次图\(S_{10}\)、\(S_{12}\)，证明对于有完美匹配及任意的三次多重图\(G\)，分别存在映射到\(E(S_{12})\)、\(E(S_{10})\)，使\(|V(f)|\)满足相应不等式，改进了已有界。
A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well-known Petersen graph, admits a 2-bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection. The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are kn…
Given a bridgeless graph $G$, the Cycle Double Cover Conjecture posits that there is a list of cycles of $G$, such that every edge appears in exactly two cycles on the list. This conjecture was originally posed independently in 1973 by Szekeres and 1979 by Seymour. In 1985, Jaeger demonstrated that it is sufficient to prove in the case that $G$ is a cubic graph. We here present a proof that every bridgeless cubic graph has a cycle double cover by analyzing certain kinds of cycles in the line graph of $G$. Further, in the case that $G$ is cubic, we prove the stronger conjecture that given a bri…
We analyse a polyhedron which contains the convex hull of all Hamiltonian cycles of a given undirected connected cubic graph. Our constructed polyhedron is defined by polynomially-many linear constraints in polynomially-many continuous (relaxed) variables. Clearly, the emptiness of the constructed polyhedron implies that the graph is non-Hamiltonian. However, whenever a constructed polyhedron is non-empty, the result is inconclusive. Hence, the following natural question arises: if we assume that a non-empty polyhedron implies Hamiltonicity, how frequently is this diagnosis incorrect? We prove…
Cubic graphs with edges in exactly one perfect matching
本文研究非双重覆盖三次图（含恰好属于一个完美匹配的孤独边），将问题归约到3-连通三次图子类并给出归纳刻画，证明孤独边数k>6时子类为空，并对3≤k≤6给出完整刻画。
Small snarks with large oddness
在给定奇性和循环连通度的条件下，我们估计了一个三次图的最少顶点数。证明了具有不同于Petersen图的奇度$ω(G)$的无桥三次图$G$至少有$5.41\cdotω(G)$顶点，并对每个具有$2\le k\le 6$的整数$k$构造了一类具有循环连通度$k$和小奇度比$|V(G)|/ω(G)$的三次图.特别是，对于循环连通性2，4，5，和6，我们提高了上界的奇数比的snarks到7.5，13，25，和99从已知的值9，15，76，和118，分别。此外，我们还构造了一个周长为5，44个顶点上的奇数为4的循环4-连通蛇，改进了以前的最佳值46。
A remark on Petersen coloring conjecture of Jaeger
如果$G$和$H$是两个三次图，则我们记为$H\prec G$，如果$G$允许一个具有$H$的边的适当边染色$f$，使得对于$G$的每个顶点$x$，都有一个具有$f(\partial_G(x))=\partial_H(y)$的$H$的顶点$y$.设$P$和$S$分别是Petersen图和Sylvester图。本文介绍了Sylvester着色猜想。此外，我们还证明了如果$G$是一个连通无桥三次图，且有$G\prec P$，则有$G=P$ .最后，如果$G$是一个连通三次图，且有$G\prec S$，则有$G=S$ .
Some snarks are worse than others
本文研究不能由四个完美匹配覆盖边集的无桥三次图族 ${\cal S}_{\geq 5}$，通过完美匹配与 2-因子添加参数将其按复杂度分层，并关联多个经典图论猜想。
Approximate cycle double cover
本文研究圈双覆盖猜想，通过给出三次图嵌入中奇异边最小数量的非平凡上界，并设计高效算法寻找满足这些上界的嵌入，从而推进了对该猜想的近似刻画。
On the existence of graphs which can colour every regular graph
本文研究能为所有正则图着色的图的存在性，推广 Petersen 着色猜想相关结果，证明无度数限制时仅有 $S_{4}$ 能为所有无桥三次图着色，并证明对度数大于 3 的正则图不存在这样的连通图。
Cubic graphs of colouring defect 3 and conjectures of Berge and Alon-Tarsi
本研究刻画了着色缺陷为3且完美匹配指数至少为5的三次图结构，并证明此类图及特定snark存在长度至多4/3·m+1的圈覆盖，支持Berge与Alon-Tarsi猜想。
Intersecting 1-factors and nowhere-zero 5-flows
设$G$是无桥三次图，$μ_2(G)$是使$G$的两个1-因子相交于$k$边的最小数$k$.一个循环边连通三次图$n$有无零5-流，如果（1）$n \geq 6$和$μ_2(G) \leq 2$或（2）如果$n \geq 5 μ_2(G)-3$
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