Thomas Bellitto, Tereza Klimošová, Martin Merker, Marcin Witkowski, Yelena Yuditsky
We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.
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A well-known conjecture of Grünbaum and Nash-Williams proposes that 4-connected toroidal graphs are hamiltonian. The corresponding results for 4-connected planar and projective-planar graphs were proved by Tutte and by Thomas and Yu, respectively, using induction arguments that proved a stronger result, that every edge is on a hamilton cycle. However, this stronger property does not hold for 4-connected toroidal graphs: Thomassen constructed counterexamples. Thus, the standard inductive approach will not work for the torus. One possible way to modify it is by characterizing the situations wher…
Bound vertices of longest paths between two vertices in cubic graphs
本文针对Zhan关于k-连通图中两顶点间最长路径含至少k-1个内部绑定顶点的猜想，证明其对2-连通立方图成立，还证明相邻顶点情形下对3-连通立方图成立，进而推导出3-连通立方图最长圈至少有两条弦，推广并加强了Thomassen的相关结果。
Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring \emph{crumby}. Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky \cite{counter} constructed an infinite family refuting the above conjecture. Their …
A spanning tree without a vertex of degree two is called a Hist which is an abbreviation for homeomorphically irreducible spanning tree. We provide a necessary condition for the existence of a Hist in a cubic graph. As one consequence, we answer affirmatively an open question on Hists by Albertson, Berman, Hutchinson and Thomassen.
A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Many hypohamiltonian planar cubic graphs have been found, starting with constructions of Thomassen in 1981. However, all the examples found until now had 4-cycles. In this note we present the first examples of hypohamiltonian planar cubic graphs with cyclic connectivity five, and thus girth five. We show by computer search that the smallest members of this class are three graphs with 76 vertices.
Merker conjectured that if $k \ge 2$ is an integer and $G$ a 3-connected cubic planar graph of circumference at least $k$, then the set of cycle lengths of $G$ must contain at least one element of the interval $[k, 2k+2]$. We here prove that for every even integer $k \ge 6$ there is an infinite family of counterexamples.
On a conjecture of Faudree and Schelp
该研究针对Faudree和Schelp关于哈密顿连通图路径长度的猜想，构造了三次平面反例补充Thomassen的结果，还构造了圈谱前半段有大量间隙的无穷哈密顿连通图族。
Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. We prove the conjecture for Generalized Petersen graphs. We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.
Putting Tutte's counterexample to Tait's conjecture in perspective to hamiltonicity and non-hamiltonicity in certain planar cubic graphs
本文通过棱柱图和Tutte碎片构造无限家族的哈密顿图与非哈密顿图，其中Tutte的反例作为最小元素出现，同时包含Holton和McKay对Tait猜想的最小边数反例的推广。
Every Minimal Counterexample to the Erdős-Gyárfás Conjecture is Predominantly Cubic
研究 Erdős-Gyárfás 猜想的最小反例结构，证明其顶点中至少 4/7 的度数为 3，且每个顶点邻接一个度数为 3 的顶点。
A counterexample to the zero forcing versus independence conjecture for cubic and subcubic graphs
研究关于三次和次三次图的零强制与独立数猜想，通过构造特定连通图作为反例，表明Z <= alpha + 1不是连通三次图通用界，且存在Z = alpha + 2的情况，反驳了相关猜想。
Chords of longest cycles in graphs with large circumferences
本文证明了对于具有大周长的图，Thomassen 和 Harvey 关于最长圈弦的猜想均成立，并推广至包含线性森林的最长圈情形。
Let v(G) and dom(G) denote the number of vertices and the domination number of a graph G, and let 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 …
A 60-Vertex Lower Bound for Cubic Bipartite Counterexamples to the Erdős-Gyárfás Conjecture
本研究通过穷举计算将Erdős-Gyárfás猜想的立方二分反例的顶点下界从30提升至60，验证了顶点数≤58的简单立方二分图必含4、8或16环，证明过程结合Moore界、Levi图转化与受限增长搜索，结果经多重方式验证。
In 2006, Barat and Thomassen conjectured in 2006 that the edges of every planar 4-regular 4-edge-connected graph can be decomposed into copies of the star with 3 leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Using the small subgraph conditioning method of Robinson and Wormald, we prove that a random 4-regular graph has an $S_3$-decomposition asymptotically almost surely, provided the number of vertices is divisible by 3.
A strong edge-coloring $φ$ of a graph $G$ assigns colors to edges of $G$ such that $φ(e_1)\ne φ(e_2)$ whenever $e_1$ and $e_2$ are at distance no more than 1. It is equivalent to a proper vertex coloring of the square of the line graph of $G$. In 1990 Faudree, Schelp, Gyárfás, and Tuza conjectured that if $G$ is a bipartite graph with maximum degree 3 and sufficiently large girth, then $G$ has a strong edge-coloring with at most 5 colors. In 2021 this conjecture was disproved by Lužar, Mačajová, Škoviera, and Soták. Here we give an alternative construction to disprove the conjecture.
A 50-Vertex Cubic Counterexample to the Domination-versus-Edge-Domination Conjecture
本文通过一个50顶点三次图反例，证明Baste等人关于正则图支配数不超过边支配数的猜想不成立，并给出精确计算与验证。
Counterexamples to two conjectures on (1, 2)-domination in cubic graphs
本文针对立方图中(1,2)-控制的两个猜想，找到非三叶虫图的反例并构造无限反例族，证明两猜想不成立，不影响Henning等人的相关猜想。
Two-coloring cubic graphs with small monochromatic components, but without singletons
本文结合小单色分量与无孤立点两个染色性质，证明存在常数c使每个三次图可2-染色，使每个单色分量顶点数介于2和c之间，并给出非平衡版本，作为Wegner猜想工具的松弛。
Subdivided expanders and counterexamples to the Tree Product Conjecture
本文为树积猜想的所有d≥2情形提供反例，通过构造具有度d多项式增长的三次扩张器细分图，推翻了该猜想，仅留d=1为开放情形。
A symmetric counterexample to Strang's conjecture for bivariate $C^1$ cubic splines on triangulations
本文构造了一个等边三角形的对称三角剖分，证明其上$C^1$三次样条空间的维数超过Strang猜想公式，给出了该猜想的反例。
Disproving the Petersen Coloring Conjecture: Theoretical Analysis and an Infinite Family of Counterexamples
针对Putman提出的112顶点彼得森着色猜想反例，本文给出不依赖大规模SAT计算的人类可验证证明，简化矛盾为彼得森图线图的结构性质，深化了对构造小装置的理解。
The Barat-Thomassen conjecture, recently proved in [Bensmail et al.: A proof of the Barat-Thomassen conjecture. J. Combin. Theory Ser. B, 124:39-55, 2017.], asserts that for every tree T, there is a constant $c_T$ such that every $c_T$-edge connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T-decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For instance, it was shown in [Bensmail et al.: Edge-partitioning a graph into paths: beyond the Barat-T…
In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen ("On the number of longest and almost longest cycles in cubic graphs", Ars Combin., 104, 307--320, 2012) about extremal graphs in these families. In particular, we find families of cyclically $5$-edge connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs $P(2n,2)$. The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes --- more precisely, the…
It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Δ^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain se…
A counterexample to the $S_{10}$- and the $S_{12}$-Conjecture
本文针对图论中受Petersen染色猜想启发的$S_{10}$-染色和$S_{12}$-染色猜想，提出了一个反例以证明这两个猜想均不成立。
Pairwise disjoint perfect matchings in $r$-edge-connected $r$-regular graphs
本文否定了Thomassen关于偶数阶$r$-边连通$r$-正则图存在$r-2$个两两不相交完美匹配的猜想（当$r \equiv 2 \text{ mod } 4$时），结合已有结果解决了所有偶数$r$的情况，并将5-边连通5-正则图的相关命题与三次图的著名猜想联系起来。
A 112-Vertex Counterexample to the Petersen Coloring Conjecture
本文构造了112顶点的无桥三次图作为彼得森着色猜想的反例，经SAT求解器验证其无彼得森着色，结合相关定理可推出无穷多此类图无该着色。
In 2006 Bar{á}t and Thomassen conjectured that every planar $4$-edge-connected $4$-regular simple graph of size divisible by three admits a claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of planar graphs with edge-connectivity $4$ which the smallest one contains $24$ vertices. In this note, we first give a smaller counterexample having only $18$ vertices and next construct a family of planar $4$-connected essentially $6$-edge-connected $4$-regular simple graphs of size divisible by three with no claw-decompositions. This result provides the sharpness for two known …
We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number $k \ge 0$ of hamiltonian cycles, which is especially efficient for small $k$. Our main findings, combining applications of this algorithm and existing algorithms with new theoretical results, revolve around graphs containing exactly one hamiltonian cycle (1H) or exactly three hamiltonian cycles (3H). Motivated by a classic result of Smith and recent work of Royle, we show that there exist nearly cubic 1H graphs of order $n$ iff $n \ge 18$ is even. This gives the strongest form of a theorem of En…
A graph $G$ admiting a $2$-factor is \textit{pseudo $2$-factor isomorphic} if the parity of the number of cycles in all its $2$-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo $2$-factor isomorphic regular bipartite graphs. Journal of Combinatorial Theory, Series B, 98(2) (2008), 432-444.] some of the authors of this note gave a partial characterisation of pseudo $2$-factor isomorphic bipartite cubic graphs and conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially $4$-edge-connected ones. In [J. Goedgebeur. A co…
A Counterexample to Wormald's Conjecture
本文构造了Wormald猜想的一个反例：一个16顶点三次图（$K_{3,3}$与两个细分$K_4$的桥接图的不交并）无法将边划分为两个同构生成线性森林，并推广至所有$16+4t$阶，已在Lean中验证。
A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph. In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional to the order of these graphs), thus provi…
We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms.
In 1971, Tutte wrote in an article that "it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian". Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts -- even if they have …
A graph $G$ is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of $G$. Abreu et al. conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs (Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture 3.6). Using a computer search we show that this conjecture is false by constructing a counterexample with 30 vertices. We also show that this is the only counterexample up to at least 40 vertices. A graph $G$ is 2-factor …
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。
A Counterexample to a Conjecture of Lovász
本文构造 r=3 时 Lovász 超图猜想的反例，即具有 102 个顶点的高度对称立方图 Biggs-Smith 图的线超图，证明该猜想不成立。
A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic graphs falls into two classes: those that can be covered with four perfect matchings, and those that need at least five. Cubic graphs that require more than four perfect matchings to cover their edges are particularly interesting as potential counterexamples to several profound and long-standing conjectures including the celebrated cycle double cover conjec…
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochstättler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochstättler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.
A 3-regular counterexample to the Bilu--Linial signing conjecture
构造一个三次图，其任意符号赋值的邻接矩阵都有特征值超出 $[-2\sqrt2,2\sqrt2]$，从而否证 Bilu--Linial 猜想对一般正则图成立，证明基于初等计算。
A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges so that every pair of distinct vertices of $G$ are connected by at least one proper path in $G$. In this paper, we consider two conjectures on the proper connection number of graphs. The first conjecture states that if $G$ is a noncomplete graph with connectivity $κ(G) = 2$ and minimum degree $δ(G)\ge 3$, then $pc(G) = 2$, posed by Borozan et al…
The Gray graph is pseudo 2-factor isomorphic
本文证明Gray图是伪2-因子同构猜想的反例，该图有54个顶点和8的圈长，是目前已知的第二大反例。通过计算机搜索验证了在特定圈长和顶点数范围内的反例不存在。
The List Total Colouring Conjecture is False
本文构造了一个三次图，其全色数为四但列表全色数为五，从而否证了列表全染色猜想，该反例由ChatGPT 6 Astra Ultra在极少人工输入下发现。
Reduction Operations and Characterizations of $S^1$-Flows in Graphs
本文针对图中$S^1$-流开发归约技术，利用双端$S^1$-预流证明相关结论，并完全刻画三角连通图、含生成三角树的图的$S^1$-流存在条件。
In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.
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。
Counterexamples to the Strong Roberson Conjecture
本文构造连通图反例，反驳强罗伯森猜想及其浸入版本，证明排除特定子式或浸入的图类同态计数仍可确定目标同态数，并给出无限族显式反例。方法基于模等价与素数阶自同构。
Log-concavity of elementary coefficients for low-rank abelian Hessenberg graphs, with a counterexample in general
研究色拟对称函数的\(e -\)对数凹性，通过展示一个\(13\)顶点连通自然单位区间图，给出其色拟对称函数的相关系数，反驳了Abreu - Nigro猜想5.3等，计算经独立确定性验证器用精确运算认证。
Small planar hypohamiltonian graphs
本文通过改造特殊子图构造出34顶点平面亚哈密顿图，并给出37、40和41顶点相关新图，改进最小阶数、三次顶点数、交叉数及自同构结果。
Milgram constructed a 28-vertex cubic graph of genus 4 that disproved Duke's conjecture relating Betti number to minimum genus. We apply Milgram's method to construct to find graphs of higher genus violating Duke's conjecture, which gives a sharper bound on that relationship. These graphs are also counterexamples to a related conjecture of Nordhaus et al. on the relationship between minimum and maximum genera of graphs. As a side note, we give a simpler proof of correctness for Milgram's method and we show that Duke's conjecture is true for genus at most 3.
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$. We show that if $|R| < \frac{3}{2}n$ and $P \cup R$ is contained in a cubic curve $c$ in the plane, then $P$ has a special property with respect to the natural group action on $c$. That is, $P$ is contained in a coset of a subgroup $H$ of $c$ of cardinality at most $|R|$. We use the same approach to show a similar result in the case where each of $B$ and $G$ is a set of $n$ points in general position …
Three-cuts are a charm: acyclicity in 3-connected cubic graphs
针对Mazzuoccolo 2013年提出的三个猜想中更强的第二个猜想（存在两个完美匹配的并的补集为无环子图），证明其虽有2-割反例，但在循环3-边连通立方图中成立。
Algorithms for the Generation of Snarks
本文提出两种生成snarks的新算法，分别用于生成girth为4和至少5的snarks，并给出了顶点数至40的完整列表，还提供了强snarks的列表。
A Counterexample to Teschner's Bondage-Number Conjecture
本研究构造18顶点连通立方二分图作为反例，否定特施纳关于图的束缚数上界为3/2倍最大度的猜想，明确给出该反例的控制数、束缚数及验证细节。
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…
Order 14 is the largest order for which every 4-total coloring of every cubic graph is equitable
本文否定Stemock关于阶数小于20的立方图的4-全着色均为均匀着色的猜想，证明14是满足该性质的最大阶数，给出相关分解引理并构造反例。
The 3-decomposition conjecture is wide open. It asserts that every finite connected cubic graph can be decomposed into a spanning tree, a disjoint union of cycles, and a matching. We show that every such decomposition is derived from a homeomorphically irreducible spanning tree (HIST). This allows us to propose a novel reformulation of the 3-decomposition conjecture: the HIST-extension conjecture. We also prove that the following graphs are reducible configurations with respect to the 3-decomposition conjecture: the triangle, the K_{2,3}, the Petersen graph with one vertex removed, the claw-sq…
For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n…
We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes -- fullgen -- and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are hamiltonian and verify that the smal…
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