In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $ω$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic graphs with $ω\geq 2$. We show that if a cubic graph $G$ has no edge cut with fewer than $ {5/2} ω- 1$ edges that separates two odd cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq {5/2} ω- 1$, then $G$ has a nowhere-zero 5-flow.
For $k \ge 4$, let $Q_{2k}$ and $V_{2k}$ denote the ladder and Möbius ladder on $2k$ vertices, respectively. We prove results that build on a result by Wormald that states that any cyclically $4$-connected cubic graph other than $Q_8$ or $V_8$ is obtained from a smaller cyclically $4$-connected cubic graph by bridging a pair of non-adjacent edges. We introduce the concept of cycle spread, which generalizes the edge pair distance defined by Wormald, and show that the set of pairs of edges that needs to be considered in order to obtain all cyclically $4$-connected cubic graphs is smaller than th…
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.
In this paper we have shown without assuming the four color theorem of planar graphs that every (bridgeless) cubic planar graph has a three-edge-coloring. This is an old-conjecture due to Tait in the squeal of efforts in settling the four-color conjecture at the end of the 19th century. We have also shown the applicability of our method to another well-known three edge-coloring conjecture on cubic graphs. Namely Tutte's conjecture that "every 2-connected cubic graph with no Petersen minor is 3-edge colorable". Hence the conclusion of this paper implies another non-computer proof of the four co…
It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte's relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte's upper bound on the chromatic polynomial of planar trian…
A note on cycles in cyclically $4$-edge-connected cubic planar graphs
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Tongji University（同济大学数学科学学院，智能计算与应用教育部重点实验室，同济大学）
Let $G$ be a cyclically $5$-connected cubic graph with a $5$-edge-cut separating $G$ into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically $5$-connected cubic graph by adding three vertices, unless $G_i$ is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity $4$ from 1988.
We present a construction which shows that there is an infinite set of cyclically 4-edge connected cubic graphs on $n$ vertices with no cycle longer than $c_4 n$ for $c_4=\frac{12}{13}$, and at the same time prove that a certain natural family of cubic graphs cannot be used to lower the shortness coefficient $c_4$ to 0. The graphs we construct are snarks so we get the same upper bound for the shortness coefficient of snarks, and we prove that the constructed graphs have an oddness growing linearly with the number of vertices.
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…
A leaf matching operation on a graph consists of removing a vertex of degree~$1$ together with its neighbour from the graph. For $k\geq 0$, let $G$ be a $d$-regular cyclically $(d-1+2k)$-edge-connected graph of even order. We prove that for any given set $X$ of $d-1+k$ edges, there is no $1$-factor of $G$ avoiding $X$ if and only if either an isolated vertex can be obtained by a series of leaf matching operations in $G-X$, or $G-X$ has an independent set that contains more than half of the vertices of~$G$. To demonstrate how to check the conditions of the theorem we prove several statements on…
A cubic graph $G$ is cyclically 5-connected if $G$ is simple, 3-connected, has at least 10 vertices and for every set $F$ of edges of size at most four, at most one component of $G\backslash F$ contains circuits. We prove that if $G$ and $H$ are cyclically 5-connected cubic graphs and $H$ topologically contains $G$, then either $G$ and $H$ are isomorphic, or (modulo well-described exceptions) there exists a cyclically 5-connected cubic graph $G'$ such that $H$ topologically contains $G'$ and $G'$ is obtained from $G$ in one of the following two ways. Either $G'$ is obtained from $G$ by subdivi…
This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976.
The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect $0$ if and only if it is $3$-edge-colourable, this invariant can measure how much a cubic graph differs from a $3$-edge-colourable graph. Our aim is to examine the relationship of colouring defect to oddness, an extensively studied measure of uncolourability of cubic graphs, defined as the smallest number of odd circuits in a $2$-factor. We show that there exist cyclically $5$-edge-connected snarks (cu…
The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with circumference ratio $c(G)/|V(G)|$ bounded from above by $0.876$, $0.960$ and $0.990$, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically $4$-edge-connected cubic graph is at least $0.75$. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. Hägglu…
It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. H$\ddot{a}$gglund constructed two graphs Blowup$(K_4, C)$ and Blowup$(Prism, C_4)$. Based on these two graphs, Chen constructed infinite families of bridgeless cubic graphs $M_{0,1,2, \ldots,k-2, k-1}$ which is obtained from cyclically 4-edge-connected and having a Fulkerson-cover cubic graphs $G_0,G_1,\ldots, G_{k-1}$ by recursive process. If each $G_i$ for $1\leq i\leq k-1$ is a cyclically 4-edge-connected snarks with excessive index at…
Let $G$ be a bridgeless cubic graph, and $μ_2(G)$ the minimum number $k$ such that two 1-factors of $G$ intersect in $k$ edges. A cyclically $n$-edge-connected cubic graph $G$ has a nowhere-zero 5-flow if (1) $n \geq 6$ and $μ_2(G) \leq 2$ or (2) if $n \geq 5 μ_2(G)-3$
The theory of voltage graphs has become a standard tool in the study graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphism that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertextransitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with a…
There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called {\em snarks}, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we…
A perfect pseudo-matching M in a cubic graph G is a spanning subgraph of G such that every component of M is isomorphic to K_2 or to K_1,3. In view of snarks G with dominating cycle C, this is a natural generalization of perfect matchings since G-E(C) is a perfect pseudo-matching. Of special interest are such M where the graph G/M is planar because such G have a cycle double cover. We show that various well known classes of snarks contain planarizing perfect pseudo-matchings, and that there are at least as many snarks with planarizing perfect pseudo-matchings as there are cyclically 5-edge-con…
Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called excessive [m]-index of G in literature. The case m=n, that is a covering with perfect matchings, is known to be strictly related to an outstanding conjecture of Berge and Fulkerson. In this paper we study in some details the case m=n-1. We show how this parameter can be large for cubic graphs with low connectivity and we furnish some evidence that each cyclically 4-connected cubic graph of order 2n has ex…
We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2-factor Q. Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction G/Q. In particular, we study the case where G is the leapfrog extension (called vertex envelope in (Discrete Math., 309(14):4793-4809, 2009)) of a plane cubic graph G_0. As a consequence we prove hamiltonicity in the leapfrog extension of planar cubic cyclically 4-edge-connected bipartite graphs. This and other results of this paper establish partial solutions of Barnette's Conjecture ac…
The perfect matching index of a cubic graph $G$, denoted by $π(G)$, is the smallest number of perfect matchings that cover all the edges of $G$. According to the Berge-Fulkerson conjecture, $π(G)\le5$ for every bridgeless cubic graph~$G$. The class of graphs with $π\ge 5$ is of particular interest as many conjectures and open problems, including the famous cycle double cover conjecture, can be reduced to it. Although nontrivial examples of such graphs are very difficult to find, a few infinite families are known, all with circular flow number $Φ_c(G)=5$. It has been therefore suggested [Electr…
The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family $\cal F$ of snarks (cyclically 4-edge-connected cubic graphs of girth at least five and chromatic index four) whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs were known. It turns out that the family $\cal F$ also has interest…
A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs. In another paper, two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove t…
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 function $f:N\rightarrow N$ is sublinear, if \[\lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.\] If $A$ is an Abelian group, $G$ is a graph and $ϕ$ is an $A$-flow in $G$, then let $N(ϕ)$ be the nullity of $ϕ$, that is, the set of edges $e$ of $G$ with $ϕ(e)=0$. In this paper we show that (a) Tutte's 5-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all $3$-edge-connected cubic graphs admit a $\mathbb{Z}_5$-flow $ϕ$ (not necessarily no-where zero), such that $|N(ϕ)|\leq f(|E(G)|)$; (b) Tutte's 4-flow conjecture is equivalent to the statement that …
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 $47/30 \cdot m \approx 1.567 m$.
Three graphs related to the \EGC\, are presented. The graphs are derived from the Buckyball, the Petersen graph, and the Tutte-Coxeter graph. The first graph is a partial answer to a question posed by Heckman and Krakovski \cite{planar} in their recent work on the planar version of the conjecture. The other two graphs appear to be the smallest known cubic graphs with no $2^m$-cycles for $m \leq 4$ and for $m \leq 5$.
We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs. Further, we establish exponential growth of the number of chromatic polynomials of planar triangulations, answering a question of D. Treumann and E. Zaslow. The structure u…
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 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…
The paper characterizes $(2t+1)$-regular graphs with circular flow number $2 + \frac{2}{2t-1}$. For $t=1$ this is Tutte's characterization of cubic graphs with flow number 4. The class of cubic graphs is the only class of odd regular graphs where a flow number separates the class 1 graphs from the class 2 graphs. We finally state some conjectures and relate them to existing flow-conjectures.
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…
The presented paper studies the flow number $F(G,σ)$ of flow-admissible signed graphs $(G,σ)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,σ)$ there is a set ${\cal G}(G,σ)$ of cubic graphs such that $F(G, σ) \leq \min \{F(H,σ_H) : (H,σ_H) \in {\cal G}(G)\}$. We prove that $F(G,σ) \leq 6$ if $(G,σ)$ contains a bridge and $F(G,σ) \leq 7$ in general. We prove better bounds, if there is an element $(H,σ_H)$ of ${\cal G}(G,σ)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,σ) \leq 4$ and the b…
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.
A {\em snark} is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph $G$ is {\em odd 2-factored} if for each 2-factor F of G each cycle of F is odd. In this paper, we present a method for constructing odd 2--factored snarks. In particular, we construct two families of odd 2-factored snarks that disprove a conjecture by some of the authors. Moreover, we approach the problem of characterizing odd 2-factored snarks furnishing a partial characterization of cyclically 4-edge connected odd 2-factored snarks. Finally, we pose a new co…
For a given snark G and edge e of G, we can form a cubic graph G_e using an operation we call "edge subtraction". The number of 3-edge-colourings of G_e is 18 * ψ(G,e) for some nonnegative integer ψ(G,e). Given snarks G_1 and G_2, we can form a new snark G using techniques given by Isaacs and Kochol. In this note we give relationships between ψ(G_1,e_1), ψ(G_2,e_2), and ψ(G,e) for particular edges e_1, e_2, and e, in G_1, G_2, and G (respectively). As a consequence, if g,h,i,j,k,l are each a nonnegative integer, then there exists a cyclically 5-edge-connected snark G with an edge e such that ψ…
A subgraph (a spanning subgraph) of a graph G whose all components are 3-vertex paths is called an L-packing (respectively, an L-factor} of G. We discuss the following old PROBLEM (A. Kelmans, 1984). Is the following claim true? (C) If G is a cubic 3-connected graph, then G has an L-packing that avoids at most two vertices of G. We show, in particular, that claim (C) is equivalent to some seemingly stronger claims (see Theorem 3.1 below). For example, if G is a cubic 3-connected graph and the number of vertices of G is divisible by three, then then the following claims are equivalent: G has an…
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 …
The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one w…
We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $ω(G)$ other than the Petersen graph has at least $5.41\cdotω(G)$ vertices, and for each integer $k$ with $2\le k\le 6$ we construct an infinite family of cubic graphs with cyclic connectivity $k$ and small oddness ratio $|V(G)|/ω(G)$. In particular, for cyclic connectivity 2, 4, 5, and 6 we improve the upper bounds on the oddness ratio of snarks to 7.5, 13, 25, and 99 from the known values 9, 15, 76, and 118, respectively. In addition,…
We prove that an infinite family of semiprimitive groups are graph-restrictive. This adds to the evidence for the validity of the PSV Conjecture and increases the minimal imprimitive degree for which this conjecture is open to 12. Our result can be seen as a generalisation of the well-known theorem of Tutte on cubic graphs. The proof uses the amalgam method, adapted to this new situation.
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 permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of M is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen sub…
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.
