发表机构
Bielefeld University of Applied Sciences; Beijing Institute of Technology, Zhuhai（比勒费尔德应用科学大学; 北京理工大学珠海学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究符号图中平衡偶环的Turán问题，刻画允许所有2k-环不平衡的图，给出双覆盖与奇偶性障碍，并证明符号四环和六环的极值数下界。
AI 中文摘要
我们研究了简单符号图中平衡偶环的Turán问题，其中符号子图在切换意义下考虑。对于每个平衡二部符号图，符号Turán数与普通Turán数相差至多两倍。我们的主要结构结果涉及那些允许一种符号指派使得每个$2k$-环都是不平衡的底层图。我们通过其$2k$-环关联向量之间不存在奇依赖来刻画这些图，给出一个上同调表述，并构造任意大阶的子图极小障碍。特别地，不存在有限的禁止子图刻画。我们还给出了双覆盖中环的精确闭游走准则，并推导出一个直接的符号广度优先搜索上界。作为应用，我们证明了\\[\hex(n,C_{+4})=\left(\frac{\sqrt2}{2}+o(1)\right)n^{3/2}\\]并研究了符号六边形数$R_6(n)=\hex(n,\{C_{-3},C_{+6}\})$。我们刻画了由$R_6$计数的底层图，并将其表述为具有指定对合的普通$C_6$-自由图的极值问题。对于每个足够大的$n$，我们构造了具有$\Omega(n^{4/3})$条边的例子，并给出一个等变构造，该构造通过双覆盖转移达到由Füredi--Naor--Verstraëte下界得到的系数。最后，我们给出具有$\Omega(n^{6/5})$条边的$n$顶点$C_{+10}$-自由符号图，并使用八边形例子来说明theta-自由性作为符号指派准则的局限性。
英文摘要
We study Turán problems for balanced even cycles in simple signed graphs, where signed subgraphs are considered up to switching. For every balanced bipartite signed graph, the signed and ordinary Turán numbers differ by at most a factor of two. Our main structural results concern the underlying graphs that admit a signing in which every $2k$-cycle is unbalanced. We characterize these graphs by the absence of an odd dependence among their $2k$-cycle incidence vectors, give a cohomological formulation, and construct subgraph-minimal obstructions of arbitrarily large order. In particular, there is no finite forbidden-subgraph characterization. We also give an exact closed-walk criterion for cycles in double covers and derive a direct signed breadth-first-search upper bound. As applications, we prove \[ \hex(n,C_{+4})=\left(\frac{\sqrt2}{2}+o(1)\right)n^{3/2} \] and study the signed hexagon number $R_6(n)=\hex(n,\{C_{-3},C_{+6}\})$. We characterize the underlying graphs counted by $R_6$ and express it as an extremal problem for ordinary $C_6$-free graphs with a prescribed involution. For every sufficiently large $n$, we construct examples with $Ω(n^{4/3})$ edges, and we give an equivariant construction attaining the coefficient obtained from the Füredi--Naor--Verstraëte lower bound by double-cover transfer. Finally, we give $n$-vertex $C_{+10}$-free signed graphs with $Ω(n^{6/5})$ edges and use octagon examples to illustrate the limitations of theta-freeness as a signing criterion.
Comments18 pages
