Related papers: Bipartite divisor graphs for integer subsets

We enumerate weighted graphs with a certain upper bound condition. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that if the given graph is a…

The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E) has the same color classes X and Y as B, and two vertices x in X and y in Y are adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph…

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total…

Let $S$ be an inverse semigroup with zero and let $Z(S)^\times$ be its set of non-zero divisors with respect to the natural partial order $\le $ on $S$, that is, $a \in Z(S)^\times $ if there exists $b\in S\setminus\{0\}$ with $\omega(a, b)…

In this note we give a combinatorial characterization of all the unmixed bipartite graphs.

For a graph $G$ of order $n$ and with eigenvalues $\lambda_1\geqslant\cdots\geqslant\lambda_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|, |\lambda_{\lceil(n+1)/2\rceil}|\right\}.$ We show that…

A word-representable graph is a simple graph $G$ which can be represented by a word $w$ over the vertices of $G$ such that any two vertices are adjacent in $G$ if and only if they alternate in $w$. It is known that the class of…

A bipartite graph $G=(A, B, E)$ is said to be a biconvex bipartite graph if there exist orderings $<_A$ in $A$ and $<_B$ in $B$ such that the neighbors of every vertex in $A$ are consecutive with respect to $<_B$ and the neighbors of every…

A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we…

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…

We consider a set of cliques in any multipartite graph with two vertices in each part. Moreover, we construct a class of peculiar polytopes. Key words: multipartite graph, clique, polytope.

A digraph is strongly connected if it has a directed path from $x$ to $y$ for every ordered pair of distinct vertices $x, y$ and it is strongly $k$-connected if it has at least $k+1$ vertices and remains strongly connected when we delete…

A graph is {\em near-bipartite} if its vertex set can be partitioned into an independent set and a set that induces a forest. It is clear that near-bipartite graphs are $3$-colorable. In this note, we show that planar graphs without cycles…

Finding dense bipartite subgraphs and detecting the relations among them is an important problem for affiliation networks that arise in a range of domains, such as social network analysis, word-document clustering, the science of science,…

Let $G$ be a $q$-regular bipartite graph with bipartition $(U,V)$. It was proved by Lu, Wang, and Yan in 2020 that $G$ has a spanning subgraph $H$ such that each vertex of $U$ has degree 1 in $H$, and each vertex of $V$ has degree distinct…

Graph theory provides powerful tools for modeling concepts in number theory, leading to the introduction of graphs derived from arithmetic properties. One such structure is the divisor prime graph, $G_{Dp(n)}$. For any positive integer $n$,…

We prove that if a graph contains the complete bipartite graph $K_{134, 12}$ as an induced minor, then it contains a cycle of length at most~12 or a theta as an induced subgraph. With a longer and more technical proof, we prove that if a…

Bipartite graph tiling was studied by Zhao who gave the best possible minimum degree conditions for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s}. Let s<t be fixed positive integers. Hladk\'y and…

In this paper, we investigate odd graceful graph, odd strongly harmonious graph, bipartite graph and their relationship. We proved following results: (1) if G is odd strongly harmonious graph, then G is odd graceful graph ;(2) if G is…