A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3- and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. A postscript is added which includes new references.
The book embedding of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. A book embedding is matching if the vertices in the same page have maximum degree at most 1. The matching book thickness is the minimum number of pages in which a graph can be matching book embedded. A graph $G$ is dispersable if and only if $mbt(G)=Δ(G)$. In this paper, we prove that bipartite cubic planar graphs are dispersable.
In a dispersable book embedding, the vertices of a given graph $G$ must be ordered along a line l, called spine, and the edges of G must be drawn at different half-planes bounded by l, called pages of the book, such that: (i) no two edges of the same page cross, and (ii) the graphs induced by the edges of each page are 1-regular. The minimum number of pages needed by any dispersable book embedding of $G$ is referred to as the dispersable book thickness $dbt(G)$ of $G$. Graph $G$ is called dispersable if $dbt(G) = Δ(G)$ holds (note that $Δ(G) \leq dbt(G)$ always holds). Back in 1979, Bernhart a…
In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is 2-extendable if and only if it is not a cycle. It is known that a non-bipartite Cayley (vertex-transitive) graph is 2-extendable when it is of minimum degree at least five \cite{sun}. We next classify all 2-extendable cubic non-bipartite Cayley graphs and obtain that: a cubic non-bipartite Cayley graph with girth $g$ is 2-extendable if and only if $g\geq 4$ and it does…
The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in [European J. Combin. 95 (2021) 103328], showed that the minimum algebraic connectivity of cubic connected graphs on $2n$ vertices is $(1+o(1))\frac{π^{2}}{2n^{2}}$, which is attained on non-bipartite graphs. Motivated by the above result, we in this paper investigate the algebraic connectivity of cubic bipartite graphs. We prove that the minimum algebraic co…
A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is {\em essentially 4-edge-connected} if it does not contain nontrivial 3-cuts. A brick $G$ is {\em near-bipartite} if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered. Kothari, de Carvalho, Lucchesi and Little proved t…
A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors have the same parity of number of circuits. In \cite{ADJLS} we proved that the only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite graph of girth 4 is $K_{3,3}$, and conjectured \cite[Conjecture 3.6]{ADJLS} that the only essentially 4--edge-connected cubic bipartite graphs are $K_{3,3}$, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations $n_3$ %{\bf decide notation and how to use it in the rest of the paper} due to Martinetti (1886) in which a…
Kang and Park recently showed that every cubic (loopless) multigraph is incidence 6-choosable [On incidence choosability of cubic graphs. \emph{arXiv}, April 2018]. Equivalently, every bipartite graph obtained by subdividing once every edge of a cubic graph, is strongly 6-edge-choosable. The aim of this note is to give a shorter proof of their result by looking at the strong edge-coloring formulation of the problem.
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.
Two edge colorings of a graph are {\em edge-Kempe equivalent} if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar bipartite cubic graphs with a range of numbers of edge-Kempe equivalence classes. Techniques are developed that will be useful for analyzing other classes of graphs as well.
Separating path systems for cubic graphs and for complete bipartite graphs
Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul（南里奥格兰德联邦大学数学与统计学院）；Centro de Modelamiento Matemático, Universidad de Chile（智利大学数学建模中心）；Instituto de Matemática, Estatística e Ciência da Computação, Universidade de São Paulo（圣保罗大学数学、统计与计算机科学学院）；Instituto Nacional de Matemática Pura e Aplicada（巴西纯粹与应用数学国家研究所）
A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one …
We prove new results for approximating Graphic TSP. Specifically, we provide a polynomial-time \frac{9}{7}-approximation algorithm for cubic bipartite graphs and a (\frac{9}{7}+\frac{1}{21(k-2)})-approximation algorithm for k-regular bipartite graphs, both of which are improved approximation factors compared to previous results. Our approach involves finding a cycle cover with relatively few cycles, which we are able to do by leveraging the fact that all cycles in bipartite graphs are of even length along with our knowledge of the structure of cubic graphs.
Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the $α+n$ conjecture, by showing that for any cubic integer $α$, there is a natural number $n$ such that $α+n$ is a chromatic roo…
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…
An incidence of a graph $G$ is a pair $(u,e)$ where $u$ is a vertex of $G$ and $e$ is an edge of $G$ incident with $u$. Two incidences $(u,e)$ and $(v,f)$ of $G$ are adjacent whenever (i) $u=v$, or (ii) $e=f$, or (iii) $uv=e$ or $uv=f$. An incidence $k$-coloring of $G$ is a mapping from the set of incidences of $G$ to a set of $k$ colors such that every two adjacent incidences receive distinct colors. The notion of incidence coloring has been introduced by Brualdi and Quinn Massey (1993) from a relation to strong edge coloring, and since then, attracted by many authors. On a list version of in…
Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed that this complex is collapsible. We point out to an oversight in his proof and explain why these complexes can be the union of collapsible complexes. Also, we prove that all links in these complexes are suspensions up to homotopy. Furthermore, we extend the definition of a cubical matching complex to planar graphs that are not necessarily bipartite, and show that these complexes are either contractible or a disjoint union of contractible complexes. For a simple connected region…
In this note we prove the conjecture of \cite{HaWiWi} that every bipartite multigraph with integer edge delays admits an edge colouring with $d+1$ colours in the special case where $d=3$. A connection to the Brualdi-Ryser-Stein conjecture is discussed.
We show that any $3$-connected cubic plane graph on $n$ vertices, with all faces of size at most $6$, can be made bipartite by deleting no more than $\sqrt{(p+3t)n/5}$ edges, where $p$ and $t$ are the numbers of pentagonal and triangular faces, respectively. In particular, any such graph can be made bipartite by deleting at most $\sqrt{12n/5}$ edges. This bound is tight, and we characterise the extremal graphs. We deduce tight lower bounds on the size of a maximum cut and a maximum independent set for this class of graphs. This extends and sharpens the results of Faria, Klein and Stehlik [SIAM…
We study the possible values of the matching number among all trees with a given degree sequence as well as all bipartite graphs with a given bipartite degree sequence. For tree degree sequences, we obtain closed formulas for the possible values. For bipartite degree sequences, we show the existence of realizations with a restricted structure, which allows to derive an analogue of the Gale-Ryser Theorem characterizing bipartite degree sequences. More precisely, we show that a bipartite degree sequence has a realization with a certain matching number if and only if a cubic number of inequalitie…
Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.
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…
Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In this paper, we provide the first superlinear bound in the general case.
For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $χ'_{int}(G,X)$ the minimum $k$ such that $G$ has an $X$-interval coloring with $k$ colors. The author and Toft conjectured [Discrete Mathematics 339 (2016), 2628--2639] that there is a polynomial $P(x)$ such that if $G$ has maximum degree at most $Δ$, then $χ'_{int}(G,X) \leq P(Δ)$. In this short note, we prove this conjecture; in fact, we prove that a cubic polynomial suffices. We al…
A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called \emph{Lehman matrices}. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs} (also known as $(α,ω)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, w…
LaBRI, CNRS, University of Bordeaux（拉布里实验室，法国国家科学研究中心，波尔多大学）；Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University（布拉迪斯拉发康斯坦丁神学院数学物理与信息学院计算机科学系）
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 …
An edge set $S$ of a connected graph $G$ is called an anti-Kekulé set if $G-S$ is connected and has no perfect matchings, where $G-S$ denotes the subgraph obtained by deleting all edges in $S$ from $G$. The anti-Kekulé number of a graph $G$, denoted by $ak(G)$, is the cardinality of a smallest anti-Kekulé set of $G$. It is NP-complete to find the smallest anti-Kekulé set of a graph. In this paper, we show that the anti-Kekulé number of a 2-connected cubic graph is either 3 or 4, and the anti-Kekulé number of a connected cubic bipartite graph is always equal to 4. Furthermore, a polynomial time…
The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders $4n+2$ and $4n+4$ with $2^n$ edge-Kempe equivalence classes are presented; it is conjectured that there are no cubic graphs with more edge-Kempe equivalence classes. New families of nonplanar bipartite cubic graphs with exactly one edge-Kempe equiva…
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minim…
Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The constructed codes are $(3,3)$-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the obtained codes and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the constructed codes. Information on the L…
In [{Structural properties and decomposition of linear balanced matrices}, {\it Mathematical Programming}, 55:129--168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for balanced bipartite graphs whose maximum degree is at most 3. We in fact obtain results for more general classes, namely linear balanceable and subcubic balanceable graphs. Additionally, we prove th…
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…
A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $χ'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we pr…
A recent result of one of the authors says that every connected subcubic bipartite graph that is not isomorphic to the Heawood graph has at least one, and in fact a positive proportion of its eigenvalues in the interval [-1,1]. We construct an infinite family of connected cubic bipartite graphs which have no eigenvalues in the open interval (-1,1), thus showing that the interval [-1,1] cannot be replaced by any smaller symmetric subinterval even when allowing any finite number of exceptions. Similar examples with vertices of larger degrees are considered and it is also shown that their eigenva…
Given a positive-weighted simple connected graph with $m$ vertices, labelled by the numbers $1,\ldots,m$, we can construct an $m \times m$ matrix whose entry $(i,j)$, for any $i,j\in\{1,\dots,m\}$, is the minimal weight of a path between $i$ and $j$, where the weight of a path is the sum of the weights of its edges. Such a matrix is called the distance matrix of the weighted graph. There is wide literature about distance matrices of weighted graphs. In this paper we characterize distance matrices of positive-weighted $n$-hypercube graphs. Moreover we show that a connected bipartite $n$-regular…
We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph. Hopkins and Staton and Bondy and Locke proved that every (sub)cubic graph of girth at least 4 has an edge-cut containing at least 4/5 of the edges. The existence of such an edge-cut follows immediately from the existence of a 5-edge-coloring as described above, so our theorem may be viewed as a coloring extension of t…
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 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 …
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…
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 2-block intersection graph (2-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements. The 2-BIGs are themselves interesting graphs: each component is cubic and 3-connected, and a 2-BIG is bipartite exactly when the TTS is decomposable to two Steiner triple systems. Any connected bipartite 2-BIG with no Hamilton cycle is a counter-example to a conjecture posed by Tutte in 1971. Our main result is that there exists an integer $N$ such that for…
