For a graph \(H\), the \(k\)-color Ramsey number \(r(H;k)\) is the least integer \(N\) such that every \(k\)-edge-coloring of \(K_N\) contains a monochromatic copy of \(H\). A \(k\)-list assignment has \(|L(e)|=k\) for every edge. The list Ramsey number \(r_\ell(H;k)\) is the least integer \(N\) for which there exists a \(k\)-list assignment on \(E(K_N)\) such that every coloring from the lists contains a monochromatic copy of \(H\). Let \(K_{1,n}\) be a star, \(S(n,m)\) the double star obtained by joining the centers of \(K_{1,n}\) and \(K_{1,m}\), and \(S_n^m\) the graph obtained from \(K_{1,n}\) by subdividing \(m\) edges once. Alon et al.\ conjectured that \(r_\ell(K_{1,n};k)=r(K_{1,n};k)\) for all \(k,n\ge1\). In this paper, we confirm their conjecture for all \(k\ge1\) and \(n\ge3\) by a unified direct proof. For even \(k\ge4\) and under explicit parameter conditions, we prove that \(r(S(n,m);k)=kn+m+2\) for even \(n\) and \(r(S_n^m;k)=k(n-1)+m+2\) for odd \(n\). For two colors, we establish a general list Ramsey lower bound and determine the common values of Ramsey and list Ramsey numbers for double and subdivided stars in explicit parameter ranges. These results close several gaps in the known bounds.
Let \(G\) be a graph with \(m\) edges and adjacency spectral radius\(\rho(G)\), and let \(N(C_{2k+1},G)\) denote the number of copies of \(C_{2k+1}\) in \(G\). For each fixed integer \(k\ge 2\), define \( g_k(m):=\frac{k-1+\sqrt{4m-k^2+1}}{2}. \) Li, Zhai and Shu [European J. Combin., 2024] determined the spectral extr...
Let \(G=C(d_1,\ldots,d_n)=F_1\BoxProd\cdots\BoxProd F_n\) be a bipartite generalized hypercube with \(n\geq2\), all \(d_i\) even, and \(N=|V(G)|\geq8\), where \(F_i=K_2\) when \(d_i=2\), and \(F_i=C_{d_i}\) when \(d_i\geq4\). We prove the following exact strengthening of two-disjoint-cycle-cover vertex bipancyclicity....
For a connected graph \(G\), let \[ Q(G)=\frac{T(G;2,0)T(G;0,2)}{T(G;1,1)^2}. \] We obtain quantitative lower bounds for \(Q\) under the graph join operation. If \(A\) and \(B\) are arbitrary simple graphs of orders \(3\le a\le b\), then \(Q(A\vee B)\) admits an explicit lower bound depending only on \(a\) and \(b\), a...
Let \(b(M)\), \(d(M)\), and \(r(M)\) denote the number of bases, the number of circuits, and the rank of a matroid \(M\), respectively. We prove that every nonempty simple binary matroid with no coloops satisfies \[ 2b(M)\ge(r(M)+1)d(M), \] with equality if and only if \(M\) is isomorphic to the Fano matroid. This conf...
For \(n\ge3\) and a graph \(H\) on at most \(n\) vertices, let \(q(H)\) be the expectation threshold for its containment in \(G(n,p)\). We prove that there are absolute constants \(a,L>0\) such that, for every \(C>0\), every graph of degeneracy at most \(C\log n/\log\log n\) has a deterministic edge decomposition into...
For a graph \(F\), the Tur\'an number \(\ex(n,F)\) is the maximum number of edges in an \(F\)-free graph on \(n\) vertices. Let \(q\ge 2\) be an integer and set \(n=q^{2}+q+1\). Brown and Erd\H{o}s, R\'enyi and S\'os independently proved that $\ex(n,C_{4})\ge \frac12 q(q+1)^{2}$ for every prime power \(q\), and F\"ured...
