Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_α, V_β\}$. We prove that for every vertex $v \in V_β$ and for every $2$-colouring $V_α \rightarrow \{1, 2\}$ there exists a $2$-colouring $V_β \rightarrow \{1, 2\}$ such that every cycle in $G$ is not monochromatic and $b(v) = 1$ ($b(v) = 2$).
Let now $G$ be a simple even plane triangulation with a vertex $3$-partition $\{V_{1}, V_{2}, V_{3}\}$. Denote by $B_{i}$, $i = 1, 2, 3$, the set of all vertices in $V_i$ of degree at least $6$ in $G$. Suppose that $G[B_{1}\cup B_{3}]$ ($G[B_{2}\cup B_{3}]$) is a subgraph of $G$ induced by the set $B_{1}\cup B_{3}$ ($B_{2}\cup B_{3}$, respectively). Let $G^{*}$ be the dual graph of $G$ with the following $3$-face-colouring: a face $f$ of $G^{*}$ is coloured with $i$ if and only if the vertex $v = f^{*} \in V_{i}$. We prove that if $H = G[B_{1}\cup B_{3}] \cup G[B_{2}\cup B_{3}] \in {\cal H}$, then, for any edge chosen on a face coloured $3$ and of size at least $6$ in $G^{*}$, there exists a Hamilton cycle of $G^{*}$ which avoids this edge. Moreover, if every component of $H$ is $2$-connected, then there exists a Hamilton cycle of $G^{*}$ such that for every face coloured $3$ it avoids every second edge of this face or it avoids at most two edges of this face.
Barnette's conjecture is an unsolved problem in graph theory. The problem states that every 3-regular (cubic), 3-connected, planar, bipartite (Barnette) graph is Hamiltonian. Partial results have been derived with restrictions on number of vertices, several properties of face-partitions and dual graphs of Barnette graphs while some studies focus just on structural characterizations of Barnette graphs. Noting that Spider web graphs are a subclass of Annular Decomposable Barnette (ADB graphs) graphs and are Hamiltonian, we study ADB graphs and their annular-connected subclass (ADB-AC graphs). We…
Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar graphs with faces of size at most 6 are Hamiltonian. We prove the conjecture.
This report provides an overview of theorems and statements related to a conjecture stated by D.W. Barnette in 1969 (which is an open problem in graph theory): Every cubic, bipartite, polyhedral graph contains a Hamilton cycle.
Let $P$ be a cubic $3$-connected bipartite plane graph which has a $2$-factor which consists only of facial $4$-cycles, and suppose that $P^{*}$ is the dual graph. We show that $P$ has at least $3^{\frac{2|P^{*}|}{Δ^{2}{(P^{*})}}}$ different Hamilton cycles.
In this paper, the concept of cyclic subsets in graph theory is introduced. An interesting theorem which relates to the collective Hamiltonicity of these cyclic subsets in graphs is also presented. This paper uses this theorem to construct an inductive proof of Barnette's long-standing conjecture, which asks whether every cubic, polyhedral, bipartite graph is Hamiltonian. Finding a class of graphs that are certain to be Hamiltonian is one of the biggest unsolved problems in Hamiltonian graph theory today.
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…
We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let $G$ be a planar triangulation. Then the dual $G^*$ is a cubic 3-connected planar graph, and $G^*$ is bipartite if and only if $G$ is Eulerian. We prove that if the vertices of $G$ are (improperly) coloured blue and red, such that the blue vertices cover the faces of $G$, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then $G^*$ is Hamiltonian. This result implies the following special case of…
A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits a partition of its vertex set into two subsets so that each induces a tree. Let $G$ be a simple even plane triangulation and suppose that ${V_1, V_2, V_3}$ is a 3-coloring of the vertex set of $G$. Let $B_{i}$, $i = 1, 2, 3$, be the set of all vertices in $V_i$ of the degree at least 6. We prove that if induced graphs $G[B_1 \cup B_2]$ and $G[B_1 \cup B_3]$ are acyclic, then the following properties are s…
Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduc…
A perfect matching cut is a perfect matching that is also a cutset, or equivalently a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22] but its complexity was open in planar graphs and in cubic graphs. We settle both questions at once by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-…
Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. Goodey proved the conjecture for the intersection of the two classes. We examine these classes from the point of view of distance-two colorings. A distance-two $r$-coloring of a graph $G$ is an assignment of $r$ colors to the vertices of $G$ so that any two vertices at distance at most two have different colors. Note that a cubic graph needs at least four colors. The distance-two four-coloring problem for cubic planar graphs is known to be NP-complete. We claim …
Barnette conjectured that all cubic $3$-connected plane graphs with maximum face size at most $6$ are hamiltonian. We provide a method of construction of a hamiltonian cycle (in dual terms) in an arbitrary cubic, $3$-connected plane graph possessing such a face $g$ that every face incident with $g$ has at most $5$ edges and every other face has at most $6$ edges.
It is a well-known fact that hamiltonicity in planar cubic graphs is an NP-complete problem. This implies that the existence of an A-trail in plane eulerian graphs is also an NP-complete problem even if restricted to planar 3-connected eulerian graphs. In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. We show, in particular, that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite grap…
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 …
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.
The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle. We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture.
Given a bridgeless graph G, the well-known cycle double cover conjecture posits that there is a list of cycles of G, such that every edge appears in exactly two cycles. In this paper, we prove the cycle double cover conjecture. More precisely, we prove the Goddyn's conjecture as a stronger version of the cycle double cover conjecture which states that every cycle in G is a member of some cycle double cover of G.
Hajós' conjecture states that an Eulerian graph of order n can be decomposed into at most (n-1)/2 edge-disjoint cycles. We describe preprocessing steps, heuristics and integer programming techniques that enable us to verify Hajós' conjecture for all Eulerian graphs with up to twelve nodes.
In 2010, N. Lichiardopol conjectured for $q \geq 3$ and $k \geq 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. We prove this conjecture for $q \geq 5$. Since it is already known to hold for $q\le4$, this completes the proof of the conjecture.
Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each non-directed cycle of order m, 2 < m < n+1.
It has been conjectured that for any fixed $r\geq 2$ and sufficiently large $n$, there is a monochromatic Hamiltonian Berge-cycle in every $(r-1)$-coloring of the edges of $K_{n}^{r}$, the complete $r$-uniform hypergraph on $n$ vertices. In this paper we prove this conjecture.
Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some variants significantly improving the result expected by the given hypothesis. Similarly, two new lower bounds for the circumference (the length of a longest cycle) are established for the reverse hypothesis proposed by Jung (2001).
In this paper, we prove the following conjecture proposed by Gould, Hirohata and Keller [Discrete Math. submitted]: Let $G$ be a graph of sufficiently large order. If $σ_t(G) \geq 2kt - t + 1$ for any two integers $k \geq 2$ and $t \geq 5$, then $G$ contains $k$ disjoint cycles.
In this paper, a proof of the cycle double cover conjecture is presented. The cycle double cover conjecture purports that if a graph is bridgeless, then there exists a list of cycles in the graph such that every edge in the graph appears in the list exactly twice. By applying induction on the number of edges in a bridgeless graph, I show that when an edge is added to a bridgeless graph, we can reform the cycle double cover to include that edge. By mathematical induction, this concludes the general CDC.
A linear cycle in a hypergraph $H$ is a cyclic sequence of hyperedges such that two consecutive hyperedges intersect in exactly one element and two nonconsecutive hyperedges are disjoint and $α(H)$ denotes the size of a largest independent set of $H$. In this note, we show that the vertex set of every $3$-uniform hypergraph $H$ can be covered by at most $α(H)$ pairwise edge-disjoint linear cycles (where we accept a vertex and a hyperedge as a linear cycle), proving a weaker version of a conjecture of Gyárfás and Sárközy.
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.
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…
Hajós conjecture asserts that a simple Eulerian graph on n vertices can be decomposed into at most (n - 1)/2 cycles. The conjecture is only proved for graph classes in which every element contains vertices of degree 2 or 4. We develop new techniques to construct cycle decompositions. They work on the common neighbourhood of two degree-6 vertices. With these techniques we find structures that cannot occur in a minimal counterexample to Hajós conjecture and verify the conjecture for Eulerian graphs of pathwidth at most 6. This implies that these graphs satisfy the small cycle double cover conjec…
Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Cavers et al. Skew-adjacency matrices of graphs, Linear Algebra Appl. 436(2012), 4512--1829] showed that the spectral radius of $G^σ$ is the same for every orientation $σ$ of $G$, and equals the maximum matching root of $G$. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ are isomorphic to the odd-cycle graph with one vertex degree $n-1$ and size $m=\lfloor 3(n-1)/2\rfloor$. This paper, by using the Kelmans transformation, …
In 1992, Manoussakis conjectured that a strongly 2-connected digraph $D$ on $n$ vertices is hamiltonian if for every two distinct pairs of independent vertices $x,y$ and $w,z$ we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. In this note we show that $D$ has a Hamilton path, which gives an affirmative evidence supporting this conjecture.
A conjecture of Birmelé, Bondy and Reed states that for any integer $\ell\geq 3$, every graph $G$ without two vertex-disjoint cycles of length at least $\ell$ contains a set of at most $\ell$ vertices which meets all cycles of length at least $\ell$. They showed the existence of such a set of at most $2\ell+3$ vertices. This was improved by Meierling, Rautenbach and Sasse to $5\ell/3+29/2$. Here we present a proof showing that at most $3\ell/2+7/2$ vertices suffice.
Thomassen conjectured that there is a function $f(k)$ such that every strongly $f(k)$-connected tournament contains $k$ edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that $f(k)\leq O(k^2(\log k)^2)$ and conjectured that there is a constant $C$ such that $f(k)\leq Ck^2$. We prove this conjecture.
Answering a question of Häggkvist and Scott, Verstraëte proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that the same holds for every graph $G$ with average degree at least $k^2+3k+2$. In this paper we prove this conjecture for $k\geq 19$ when $G$ is sufficiently large. We also show that for any $ε>0$ and large $k\geq k_ε$, average degree at least $k^2+3k-2+ε$ suffices, which is asymptotically tight for infinitely many graphs.
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.
In the theory of digraphs, the study of cycles is a subject of great importance and has given birth to a number of deep questions such as the Behzad-Chartrand-Wall conjecture (1970) and its generalization, the Caccetta-Häggkvist conjecture (1978). Despite a lot of interest and efforts, the progress on these remains slow and mostly restricted to the solution of some special cases. In this note, we prove these conjectures for digraphs with girth is at least as large as their minimum out-degree and without short even cycles. More generally, we prove that if a digraph has sufficiently large girth …
In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach. (1) Every graph $G$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $k$; in addition, if $G$ is 2-connected and non-bipartite, then i…
School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
本文证明了Dean猜想对k≥6成立，并提出了更强的结果，证明每个最小度数至少为k的图中存在长度为r mod k的循环，除非存在特定例外图。
For a simple digraph $G$ without directed triangles or digons, let $β(G)$ be the size of the smallest subset $X \subseteq E(G)$ such that $G\setminus X$ has no directed cycles, and let $γ(G)$ be the number of unordered pairs of nonadjacent vertices in $G$. In 2008, Chudnovsky, Seymour, and Sullivan showed that $β(G) \le γ(G)$, and conjectured that $β(G) \le γ(G)/2$. Recently, Dunkum, Hamburger, and Pór proved that $β(G) \le 0.88 γ(G)$. In this note, we prove that $β(G) \le 0.8616 γ(G)$.
A conjecture of Thomassen from 1982 states that for every k there is an f(k) so that every strongly f(k)-connected tournament contains k edge-disjoint Hamilton cycles. A classical theorem of Camion, that every strongly connected tournament contains a Hamilton cycle, implies that f(1)=1. So far, even the existence of f(2) was open. In this paper, we prove Thomassen's conjecture by showing that f(k)=O(k^2*log^2(k)). This is best possible up to the logarithmic factor. As a tool, we show that every strongly 10^4*k*log(k)-connected tournament is k-linked (which improves a previous exponential bound…
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 $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size $k$ formed by $k$ consecutive vertices in the ordering. We prove that every red-blue edge-coloured $K_n^{(4)}$ contains a red and a blue tight cycle that are vertex-disjoint and together cover $n-o(n)$ vertices. Moreover, we prove that every red-blue edge-coloured $K_n^{(5)}$ contains four monochromatic tight cycles that are vertex-disjoint and together cover $n-o(n)$ vertices.
Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in [Bondy, Murty, Graph Theory, Springer-Verlag London, 2008]. Lichiardopol, Por and Sereni proved in [SIAM J. Discrete Math. 23 (2) (2009) 979-992] that the above conjecture holds for $k=3$. Let $g$ be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy and Thomassé conjectured in [J. Graph Theory 75 (3) (2014) 284-302] that …
A pebbling move on a graph $G$ consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number of a graph $G$, denoted by $f(G)$, is the least integer $n$ such that, however $n$ pebbles are located on the vertices of $G$, we can move one pebble to any vertex by a sequence of pebbling moves. Let $M(G)$ be the middle graph of $G$. For any connected graphs $G$ and $H$, Graham conjectured that $f(G\times H)\leq f(G)f(H)$. In this paper, we give the pebbling number of some graphs and prove that Graham's conjecture holds for the middle graphs of some even cyc…
In this article, we give a positive answer to the cycle double cover conjecture. Ones who are mainly interesting in the proof of the conjecture can only read Sections 2 and 4.
In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstraëte, who asked whether the regularity can be replaced with the weaker condition that the minimum degree is at least $3$. Despite attention from various researchers, until now, the best partial result towards both of these conjectures was a $\sqrt{n}$ lower bound on the number of cycle lengths. We resolve these conjectures asymptotically, by showing that the number of cycle lengths is at least $n^{1-o(1)}…
Bermond and Thomassen conjectured that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex disjoint cycles. So far the conjecture was verified for $k\le 3$. Here we generalise the question asking for all outdegree sequences which force $k$ vertex disjoint cycles and give the full answer for $k\le 2$.
In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: For every $c>1/2$ and $\varepsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n\geq n_0$ vertices contains $(1-\varepsilon)cn$ edge-disjoint Hamilton cycles.
