Motivated by the Ruzsa-Szemerédi problem, Imolay, Karl, Nazy, and Váli studied a variant of Turán number ex_F(n,G) (called the F-multicolor Turán number of G), defined as the maximum number of edge-disjoint copies of F on n-vertex set such that there is no copies of G whose edges come from distinct copies of F. They proved that if there is no homomorphism from G to F, then n^2/v(F)^2+o(n^2)≤ ex_F(n,G)≤ ex(n,G)/e(F)+o(n^2), and otherwise ex_F(n,G) = o(n^2). The quantity ex_F(n,G) asymptotically equals the maximum size of an F-packing in an n-vertex G-free graph, and attains the upper bound ex(n,G)/e(F)+o(n^2) if and only if χ(G) > χ(F). In this paper, we provide conditions under which ex_F(n,G) does not achieve the lower bound n^2/v(F)^2 + o(n^2), and describe additional graph pairs that attain this lower bound via graph blow-ups. Especially, we proved that ex_C_k(s)(n,C_k-2)=n^2/(sk)^2+o(n^2) for any k≥ 5. For degenerate cases, we show that if χ(F) = 3 and G and F share the same odd girth, then ex_F(n,G) satisfies the (6,3)-type bound n^2-o(1), generalizing a result of Kovács and Nagy. We also prove that ex_C_2k+1(n,C_2ℓ+1)=O(n^1+1/(ℓ-k+1)) for any integers k,ℓ with ℓ>k, extending a result of Füredi and Özkahya. Additionally, we establish ex_C_4(n,C_4)=√(2)n^3/2/8+O(n).

Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are\emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if$E(T_i) \cap E(T_j )=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for everypair of distinct integers $i,j$, $1 \leq i, j \leq \ell$. Similarly,if we only have the condition $E(T_i) \cap E(T_j )=\emptyset$but without the condition $V(T_i)\cap V(T_j)=S$, then they are \emph{edge-disjointSteiner trees}.The \emph{generalized $k$-connectivity}, denoted by $\kappa_k(G)$,of a graph $G$, is defined as $\kappa_k(G)=\min\{\kappa_G(S)|S\subseteq V(G) \ \textrm{and} \ |S|=k \}$, where $\kappa_G(S)$ isthe maximum number of internally disjoint $S$-Steiner trees. The \emph{generalized local edge-connectivity}$\lambda_{G}(S)$ is the maximum number of edge-disjoint Steiner treesconnecting $S$ in $G$. The {\it generalized $k$-edge-connectivity}$\lambda_k(G)$ of $G$ is defined as$\lambda_k(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G) \ and \ |S|=k\}$.These measures aregeneralizations of the concepts of connectivity and edge-connectivity, and they and can be used as measuresof vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs.In this paper, we obtain the exact value of $\lambda_{k}(S(n,\ell))$ for $3\leq k\leq \ell^n$, and the exact value of $\kappa_{k}(S(n,\ell))$ for $3\leq k\leq \ell$, where $S(n, \ell)$ is the Sierpi\'{n}ski graphs with order $\ell^n$. As a direct consequence, these graphs provide additional interesting examples when $\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$. We also study the some network properties of Sierpi\'{n}ski graphs.

An edge-coloring of graph G is called closed-neighborhood (resp. open-neighborhood) conflict-free edge-coloring if for every edge uv E E(G), there is a color assigned to exactly one edge among E(u) U E(v) (resp. E(u) U E(v) - {uv}). The smallest number of colors needed in any possible closed-neighborhood (resp. open-neighborhood) conflict- free edge-coloring of G, denoted chi ' CF[G] (resp. chi ' CF(G)), is called the closed-neighborhood (resp. open-neighborhood) conflict-free index of G. In this paper, we prove that decide whether chi ' CF[G] = 2 or chi ' CF(G) = 2 is NP-complete, even if G is a bipartite graph. (c) 2025 Published by Elsevier B.V.

The problem of determining the maximum number of copies of $T$ in an $H$-free graph, for any graphs $T$ and $H$, was considered by Alon and Shikhelman. This is a variant of Turán's classical extremal problem. We show lower and upper bounds for the maximum number of $s$-cliques in a graph with no disjoint copies of arbitrary graph. We also determine the maximum number of $s$-cliques in an $n$-vertex graph that does not contain a disjoint union of $k$ paths of length two when $k=2,3$, or $s\geqslant k+2$, or $n$ is sufficiently large, this partly confirms a conjecture posed by Chen, Yang, Yuan, and Zhang \cite{2024Chen113974}.

For two graphs G and H, the Gallai–Ramsey number gr_k(G:H) is defined as the minimum integer n such that any k-edge-coloring of K_n must contain either a rainbow copy of G or a monochromatic copy of H. In this paper, we obtain the exact values of gr_k(G:H) , where H is a path and G∈{K_1,3,P_4^+,P_5} is a small tree and P_4^+ is the graph consisting of P_4 with one extra edge incident with an inner vertex.

Let G={G1, horizontal ellipsis ,Gs} ${\bf{G}}=\{{G}_{1},\ldots ,{G}_{s}\}$ be a collection of not necessarily distinct n $n$-vertex graphs with the same vertex set V $V$. We use G similar to $\tilde{{\bf{G}}}$ to denote an edge-colored multigraph of G ${\bf{G}}$ with V(G similar to)=V $V(\tilde{{\bf{G}}})=V$ and E(G similar to) $E(\tilde{{\bf{G}}})$ a multiset consisting of E(G1), horizontal ellipsis ,E(Gs) $E({G}_{1}),\ldots ,E({G}_{s})$, and the edge e $e$ of G similar to $\tilde{{\bf{G}}}$ is colored by i $i$ if e is an element of E(Gi) $e\in E({G}_{i})$. A graph H $H$ is rainbow in G ${\bf{G}}$ if any two edges of H $H$ belong to different graphs of G ${\bf{G}}$. We say that G ${\bf{G}}$ is rainbow vertex-pancyclic if each vertex of V $V$ is contained in a rainbow cycle of G ${\bf{G}}$ with length l $\ell $ for every integer l is an element of[3,n] $\ell \in [3,n]$, and that G ${\bf{G}}$ is rainbow panconnected if for any pair of vertices u $u$ and v $v$ of V $V$ there exists a rainbow path of G ${\bf{G}}$ with length l $\ell $ joining u $u$ and v $v$ for every integer l is an element of[dG similar to(u,v),n-1] $\ell \in [{d}_{\tilde{{\bf{G}}}}(u,v),n-1]$. In this paper, we study the existences of rainbow spanning trees and rainbow Hamiltonian paths in G ${\bf{G}}$ under the Ore-type conditions. Moreover, we study the rainbow vertex-pancyclicity and rainbow panconnectedness, as well as the existence of rainbow cliques in G ${\bf{G}}$ under the Dirac-type conditions. We also give some examples to show the sharpness of our results.

An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$ and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph. We also prove that $mc(G)\leq m-n+4$ if $G$ is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.

A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29-44] that for sufficiently large n $n$, every 2-edge-connected simple graph G $G$ with n = divide V( G ) divide $n=| V(G)| $ and minimum degree delta( G ) >= n 5 $\delta (G)\ge \frac{n}{5}$ is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765-1776], it is shown that for any connected simple regular matroid M $M$, if every cocircuit D $D$ of M $M$ satisfies divide D divide >= max r( M ) - 5 5 , 6 $| D| \ge \max \left\{\frac{r(M)-5}{5},6\right\}$, then M $M$ is supereulerian. We prove the following. (i) Let M $M$ be a connected simple regular matroid. If every cocircuit D $D$ of M $M$ satisfies divide D divide >= max r( M ) + 1 10 , 9 $| D| \ge \max \left\{\frac{r(M)+1}{10},9\right\}$, then M $M$ is supereulerian. (ii) For any real number c $c$ with 0 < c < 1 $0\lt c\lt 1$, there exists an integer f( c ) $f(c)$ such that if every cocircuit D $D$ of a connected simple cographic matroid M $M$ satisfies divide D divide >= max{c(r( M ) + 1 ) , f( c ) } $| D| \ge \max \{c(r(M)+1),f(c)\}$, then M $M$ is supereulerian.

The concepts of monochromatic connection number mc(G) (MC-number for short) and vertex monochromatic connection number mvc(G) (MVC-number for short) of a graph G were introduced in 2011 and 2018, respectively, by Caro and Yuster and Cai et al., and have been studied extensively, While in 2017, Jiang et al. introduced the concept of total monochromatic connection number tmc(G) (TMC-number for shot) of a graph G. In this paper, we mainly study the TMC-number of a graph. At first, we completely determine the TMC-numbers for any given simple and connected graphs, and obtain some Nordhaus-Gaddum-type results for the TMC-number. Jiang et al. in 2017 put forward a conjecture and a problem on the difference between tmc(G), mc(G) and mvc(G) of a graph G. We then completely solve the conjecture and the problem, and characterize the graphs G of order n with $$tmc(G)-mc(G)=n-1$$ .

Fault-tolerant networks are often modeled as s -hamiltonian graphs. Thus it is of interests to find graph families in which whether a graph is s -hamiltonian can be determined in polynomial time. An hourglass is a graph obtained from K 5 by deleting the edges in a cycle of length 4, and an hourglass-free graph is one that has no induced subgraph isomorphic to an hourglass. Kriesell in [J. Combin. Theory Ser. B, 82 (2001), 306-315] proved that every 4-connected hourglass-free line graph is Hamilton-connected, and Kaiser, Ryjáček and Vrána in [Discrete Mathematics, 321 (2014) 1-11] extended it by showing that every 4-connected hourglass-free line graph is 1-Hamilton-connected. We characterize all essentially 4-edge-connected graphs whose line graph is hourglass-free. Consequently we prove that for any integer s and for any hourglass-free line graph L ( G ), each of the following holds. (i) If s ≥ 2, then L ( G ) is s -hamiltonian if and only if κ ( L ( G ) ) ≥ s + 2; (ii) If s ≥ 1, then L ( G ) is s -Hamilton-connected if and only if κ ( L ( G ) ) ≥ s + 3.

The concept of rainbow disconnection number of graphs was introduced by Chartrand et al. (2018). Inspired by this concept, we put forward the concepts of rainbow vertex-disconnection and proper disconnection in graphs. In this paper, we first show that it is NP-complete to decide whether a given edge-colored graph G has a proper edge-cut separating two specified vertices, even though the graph G has $$\Delta (G)=4$$ or is bipartite. Then, for a graph G with $$\Delta (G)\le 3$$ we show that $$pd(G)\le 2$$ and distinguish the graphs with $$pd(G)=1$$ and 2, respectively. We also show that it is NP-complete to decide whether a given vertex-colored graph G is rainbow vertex-disconnected, even though the graph G has $$\Delta (G)=3$$ or is bipartite.

For an edge-colored graph G, we call an edge–cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatically disconnected if any two distinct vertices of G are separated by a monochromatic edge–cut. For a connected graph G, the monochromatic disconnection number of G, denoted by md(G), is the maximum number of colors that are needed in order to make G monochromatically disconnected. We show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus–Gaddum-type results for md(G).

A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call k paths $$P_1,\ldots ,P_k$$ rainbow monochromatic paths if every $$P_i$$ is monochromatic and for any two $$i\ne j$$ , $$P_i$$ and $$P_j$$ have different colors. An edge-coloring of a graph G is said to be a rainbow monochromatic k-edge-connection coloring (or $$RMC_k$$ -coloring for short) if every two distinct vertices of G are connected by at least k rainbow monochromatic paths. We use $$rmc_k(G)$$ to denote the maximum number of colors that ensures G has an $$RMC_k$$ -coloring, and this number is called the rainbow monochromatic k-edge-connection number. We prove the existence of $$RMC_k$$ -colorings of graphs, and then give some bounds of $$rmc_k(G)$$ and present some graphs whose $$rmc_k(G)$$ reaches the lower bound. We also obtain the threshold function for $$rmc_k(G(n,p))\ge f(n)$$ , where $$\left\lfloor \frac{n}{2}\right\rfloor > k\ge 1$$ .

For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatic disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. For a connected graph G, the monochromatic disconnection number (or MD-number for short) of G, denoted by md(G), is the maximum number of colors that are allowed in order to make G monochromatic disconnected. For graphs with diameter one, they are complete graphs and so their MD-numbers are 1. For graphs with diameter at least 3, we can construct 2-connected graphs such that their MD-numbers can be arbitrarily large; whereas for graphs G with diameter two, we show that if G is a 2-connected graph then md(G)≤2, and if G has a cut-vertex then md(G) is equal to the number of blocks of G. So, we will focus on studying 2-connected graphs with diameter two, and give two upper bounds of their MD-numbers depending on their connectivity and independent numbers, respectively. We also characterize the n2-connected graphs (with large connectivity) whose MD-numbers are 2 and the 2-connected graphs (with small connectivity) whose MD-numbers achieve the upper bound n2 (these graphs are called extremal graphs). For graphs with connectivity less than n2, we show that if the connectivity of a graph is linear in its order n, then its MD-number is upper bounded by a constant, and this suggests us to leave a conjecture that for a k-connected graph G, md(G)≤nk.

For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with a same color. The graph G is called monochromatically disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. The monochromatic disconnection number, denoted by md(G), of a connected graph G is the maximum number of colors that are allowed to make G monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph products, i.e., Cartesian, strong, lexicographic, and tensor products.

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

For an edge-colored graph G, a set F of edges of G is called a proper edge-cut if F is an edge-cut of G and any pair of adjacent edges in F are assigned by different colors. An edge-colored graph is called proper disconnected if for each pair of distinct vertices of G there exists a proper edge-cut separating them. For a connected graph G, the proper disconnection number of G, denoted by pd(G), is defined as the minimum number of colors that are needed to make G proper disconnected. In this paper, we first show that it is NP-complete to decide whether a given k-edge-colored graph G with \(\varDelta (G)=4\) is proper disconnected. Then, for a graph G with \(\varDelta (G)\le 3\) we show that \(pd(G)\le 2\) and determine the graphs with \(pd(G)=1\) and 2 in polynomial time, respectively, when the set of vertices with degree 3 in G is an independent set. Finally, we show that for a general graph G, deciding whether \(pd(G)=1\) is NP-complete, even if G is bipartite.

A path in an edge-colored graph G is called monochromatic if any two edges on the path have the same color. For k≥2, an edge-colored graph G is said to be monochromatic k-edge-connected if every two distinct vertices of G are connected by at least k edge-disjoint monochromatic paths, and G is said to be uniformly monochromatic k-edge-connected if every two distinct vertices are connected by at least k edge-disjoint monochromatic paths such that all edges of these k paths are colored with a same color. We use mck(G) and umck(G) to denote the maximum number of colors that ensures G to be monochromatic k-edge-connected and, respectively, G to be uniformly monochromatic k-edge-connected. In this paper, we first conjecture that for any k-edge-connected graph G, mck(G)=e(G)−e(H)+⌊k2⌋, where H is a minimum k-edge-connected spanning subgraph of G. We verify the conjecture for k=2. We also prove the conjecture for G=Kk+1 and G=Kk,n with n≥k≥3. When G is a minimal k-edge-connected graph, we give an upper bound of mck(G), i.e., mck(G)≤k−1. For the uniformly monochromatic k-edge-connectivity, we prove that for all k, umck(G)=e(G)−e(H)+1, where H is a minimum k-edge-connected spanning subgraph of G.

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G − S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of (G − xy) − S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G)= k for given integers k and n with 1 ≤ k ≤ n.

A matroid M with a distinguished element $$e_0 \in E(M)$$ is a rooted matroid with $$e_0$$ being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely many such matroids, the number of serial reductions of such matroids is finite. In particular, we find two finite families of binary matroids $$\mathcal M_1$$ and $$\mathcal M_2$$ and prove the following. (i) For some $$e_0 \in E(M)$$ , M has at most three circuits containing $$e_0$$ if and only if the serial reduction of M is isomorphic to a member in $$\mathcal M_1$$ . (ii) If for some $$e_0 \in E(M)$$ , M has at most three circuits not containing $$e_0$$ if and only if the serial reduction of M is isomorphic to a member in $$\mathcal M_2$$ . These characterizations will be applied to show that every connected binary matroid M with at least four circuits has a 1-hamiltonian circuit graph.