Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.     

The reconstruction conjecture for finite simple graphs and associated directed graphs

In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let Γ and ${\mathrm{\Gamma }}^{\prime }$ be finite simple graphs with at least three vertices such that there exists a bijective map $f:V(\mathrm{\Gamma })\to V({\mathrm{\Gamma }}^{\prime })$ and for any $v\in V(\mathrm{\Gamma })$, there exists an isomorphism ${?}_v:\mathrm{\Gamma }?v\to {\mathrm{\Gamma }}^{\prime }?f(v)$. Then we define the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}=\stackrel{?}{\mathrm{\Gamma }}(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime },f,{\{{?}_v\}}_{v\in V(\mathrm{\Gamma })})$ with two kinds of arrows from the graphs Γ and ${\mathrm{\Gamma }}^{\prime }$, the bijective map f and the isomorphisms ${\{{?}_v\}}_{v\in V(\mathrm{\Gamma })}$. By investigating the associated directed graph $\stackrel{?}{\mathrm{\Gamma }}$, we study when are the two graphs Γ and ${\mathrm{\Gamma }}^{\prime }$ isomorphic.     

Counterexamples to the edge reconstruction conjecture for infinite graphs

For every infinite cardinal α, there exists a graph with α edges which is not uniquely reconstructible from its family of edge-deleted subgraphs.     

Gallai’s path decomposition conjecture for triangle-free planar graphs

A path decomposition of a graph $G$ is a collection of edge-disjoint paths of $G$ that covers the edge set of $G$. Gallai (1968) conjectured that every connected graph on $n$ vertices admits a path decomposition of cardinality at most $?(n+1)∕2?$. Gallai’s Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai’s Conjecture for graphs with maximum degree at most 5, and Botler et al. (2017) verified it for graphs with treewidth at most 3. In this paper, we verify Gallai’s Conjecture for triangle-free planar graphs.     

Bipartite almost distance-hereditary graphs

The notion of distance-heredity in graphs has been extended to construct the class of almost distance-hereditary graphs (an increase of the distance by one unit is allowed by induced subgraphs). These graphs have been characterized in terms of forbidden induced subgraphs [M. Aïder, Almost distance-hereditary graphs, Discrete Math. 242 (1–3) (2002) 1–16]. Since the distance in bipartite graphs cannot increase exactly by one unit, we have to adapt this notion to the bipartite case.In this paper, we define the class of bipartite almost distance-hereditary graphs (an increase of the distance by two is allowed by induced subgraphs) and obtain a characterization in terms of forbidden induced subgraphs.     

Gallai’s path decomposition conjecture for graphs of small maximum degree

Gallai’s path decomposition conjecture states that the edges of any connected graph on $n$ vertices can be decomposed into at most $\frac{n+1}{2}$ paths. We confirm that conjecture for all graphs with maximum degree at most five.     

Bipartite density of cubic graphs

We first obtain the exact value for bipartite density of a cubic line graph on n vertices. Then we give an upper bound for the bipartite density of cubic graphs in terms of the smallest eigenvalue of the adjacency matrix. In addition, we characterize, except in the case n=20, those graphs for which the upper bound is obtained.     

Bipartite graphs and their endomorphism monoids

It is shown that any connected bipartite graph is determined by its endomorphism monoid up to isomorphism. © 1996 John Wiley & Sons, Inc.     

Behzad-Vizing conjecture and Cartesian-product graphs

We prove the following theorem: if the Behzad-Vizing conjecture is true for graphs G and H, then is it true for the Cartesian product G□H.     

The reconstruction conjecture is true if all 2-connected graphs are reconstructible

It is shown that the Reconstruction Conjecture is true for all finite graphs if it is true for the 2-connected ones.     

Bipartite algebraic graphs without quadrilaterals

Let ${\mathbb{P}}^s$ be the $s$-dimensional complex projective space, and let $X,Y$ be two non-empty open subsets of ${\mathbb{P}}^s$ in the Zariski topology. A hypersurface $H$ in ${\mathbb{P}}^s\times {\mathbb{P}}^s$ induces a bipartite graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set is defined by $\overline{u}～\overline{v}$ if and only if $(\overline{u},\overline{v})\in H$. Motivated by the Turán problem for bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$ and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in $\overline{y}$ is bounded by a constant $d=d(s,t)$, and we discuss possible notions of the equivalence.We establish the result that if $H\cap (X\times {\mathbb{P}}^2)$ is $(2,2)$-grid-free, then there exists $F\in \mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such that $H\cap (X\times {\mathbb{P}}^2)=\{F=0\}\cap (X\times {\mathbb{P}}^2)$. Finally, we transfer the result to algebraically closed fields of large characteristic.     

Bipartite dimensions and bipartite degrees of graphs

A cover (bipartite) of a graph G is a family of complete bipartite subgraphs of G whose edges cover G's edges. G'sbipartite dimension d(G) is the minimum cardinality of a cover, and its bipartite degree η(G) is the minimum over all covers of the maximum number of covering members incident to a vertex. We prove that d(G) equals the Boolean interval dimension of the irreflexive complement of G, identify the 21 minimal forbidden induced subgraphs for d 2, and investigate the forbidden graphs for d n that have the fewest vertices. We note that for complete graphs, d(K_n) = [log₂n], η(K_n) = d(K_n) for n 16, and η(K_n) is unbounded. The list of minimal forbidden induced subgraphs for η 2 is infinite. We identify two infinite families in this list along with all members that have fewer than seven vertices.     

Bipartite distance-regular graphs of valency three

Bipartite distance-regular graphs of valency three are classified. There are eight such graphs, all of which have diameter less than 9, and seven of them are distance- transitive.     

Bipartite density of triangle-free subcubic graphs

A graph is subcubic if its maximum degree is at most 3. The bipartite density of a graph G is defined as b(G)=max{|E(B)|/|E(G)|:B is a bipartite subgraph of G}. It was conjectured by Bondy and Locke that if G is a triangle-free subcubic graph, then and equality holds only if G is in a list of seven small graphs. The conjecture has been confirmed recently by Xu and Yu. This note gives a shorter proof of this result.     

On multiplicative graphs and the product conjecture

We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Ne?et?il and Pultr These results allow us (in conjunction with earlier results of Ne?et?il and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].