The study of a variation the marking game, in which first player marks vertices and second edges an undirected graph was proposed by Bartnicki et al. (Electron J Combin 15:R72, 2008). In this goal is to mark as many around unmarked vertex possible, while wants just opposite. paper, we prove various bounds for corresponding invariant, vertex-edge coloring number $${\text {col}}_\mathrm{ve}(G)$$ ...

Given a graph $G$, 2-coloring of the edges $K_n$ is said to contain balanced copy $G$ if we can find such that half its in each color class. If there exists an integer $k$ that, for $n$ sufficiently large, every with more than contains then say balanceable. The smallest this holds called balancing number $G$.In paper, define general variant number, generalized by considering 2-coverings edge se...

Let H = (V, E) be a hypergraph, where V is set of vertices and E non-empty subsets called edges. If all edges have the same cardinality r, then an r-uniform hypergraph; if consists r-subsets V, complete denoted by $$K_n^r$$ , n |V|. A hypergraph H′ (V′, E′) subhypergraph V′ ⊆ E′ E. The edge-connectivity minimum edge F such that − not connected, F). An k-edge-maximal every has at most k, but for...

We study the minimum spanning tree problem on complete graph $K_n$ where an edge $e$ has a weight $W_e$ and cost $C_e$, each of which is independent copy random variable $U^\gamma$ $\gamma\leq 1$ $U$ uniform $[0,1]$ variable. There also constraint that $T$ must satisfy $C(T)\leq c_0$. establish, for range values $c_0,\gamma$, asymptotic value optimum via consideration dual problem.

We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ is said to be zero-sum under $f$ if $f(H) := \sum_{e\in E(H)} f(e) =0$. study following type questions, several cases obtaining best possible results: Under which conditions on $|f(G)|$ can we guarantee existence spanning tree $G$? The types are comple...

For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ minimum number colors needed for have q)$-coloring. This was introduced about 45 years ago, but studied systematically by Erd\H{o}s Gy\'{a}rf\'{a}s in 1997, now known as Erd\H{o}s-Gy\'{a}rf\'{a}s function. In this p...

Let $T_{\rho}$ be an irrational rotation on a unit circle $S^{1}\simeq [0,1)$. Consider the sequence $\{\mathcal{P}_{n}\}$ of increasing partitions $S^{1}$. Define hitting times $N_{n}(\mathcal{P}_n;x,y):= \inf\{j\geq 1\mid T^{j}_{\rho}(y)\in P_{n}(x)\}$, where $P_{n}(x)$ is element $\mathcal{P}_{n}$ containing $x$. D. Kim and B. Seo in [9] proved that rescaled $K_n(\mathcal{Q}_n;x,y):= \frac{\...