Related papers: A Characterization On Potentially $K_{2,5}$-graphi…

In this paper we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class $\overline G_{m,n}$ of the complements of such graphs. We establish various properties of…

A graph is a ``$k$-Kuratowski graph'' if it has exactly $k$ components, each isomorphic to $K_5$ or to $K_{3,3}$. We prove that if a graph $G$ contains no $k$-Kuratowski graph as a minor,then there is a set $X$ of boundedly many vertices…

An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered. Given $2\leq k\leq r$, an ordered $r$-graph $H$ is {\em interval} $k$-{\em partite} if there exist at least $k$ disjoint intervals in the ordering…

Let $G$ be a graph with vertex set $V(G)$ and let $H:V(G)\rightarrow 2^N$ be a set function associating with $G$. An $H$-factor of graph $G$ is a spanning subgraphs $F$ such that $$d_F(v)\in H(v){4em}\hbox{for every}v\in V(G).$$ Let…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

A graph $G$ admits an $H$-tiling if it contains a collection of vertex-disjoint copies of $H$. In this paper, we confirm a conjecture proposed by K\"{u}hn, Osthus, and Treglown by showing that for any given graph $H$, there exists a…

Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $\rho : A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0,…

Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$…

Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $\delta(H_n)\geq\alpha n$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will…

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and…

A graph $G$ is said to be a $(k,\ell)$-graph if its vertex set can be partitioned into $k$ independent sets and $\ell$ cliques. It is well established that the recognition problem for $(k,\ell)$-graphs is NP-complete whenever $k \geq 3$ or…

Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $\pi : V(H) \longrightarrow \{1,2,\ldots, k\}$ so that each color class induces a $K_2$-free subgraph. For a graph $G$ of order at least $2$, a $G$-free $k$-coloring of $H$…

For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\chi(H)> m$ the behavior…

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

The k-planar graphs, which are (usually with small values of k such as 1, 2, 3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently…

Let $P(G)=(P_{0}(G),P_{1}(G),\cdots, P_{\rho}(G))$ be the path sequence of a graph $G$, where $P_{i}(G)$ is the number of paths with length $i$ and $\rho$ is the length of a longest path in $G$. In this paper, we first give the path…

The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex…

In this paper we show that certain almost distance-regular graphs, the so-called $h$-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph $G$ with diameter $D$ is called…

A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if…