How many disjoint copies of some graph $H$ can appear in an induced subgraph of a strongly regular graph? We give upper and lower bounds that depend on the number of vertices in their boundary. This generalizes a result of Lou-Murin which bounds the independence number of a (hitherto still elusive) strongly regular graph with parameters $(99,14,1,2)$. The equality cases suggest alternative structural approaches for potential constructions of such a graph. We show, using computer searches, that one cannot impose certain additional natural conditions for some of the cases.
For a graph $G$, an odd induced subgraph of $G$ is an induced subgraph in which every vertex has odd degree (within the subgraph). Let $f_o(G)$ denote the maximum size of such a subgraph in $G$. Caro conjectured that there exists a positive constant $c$ such that $f_o(G)≥cn$ for any $n$-vertex graph without isolated ve...
Xin-Ru Yang, Qing-Hou Zeng· Annals of Applied Mathematic...· 0 citations
We prove that there is an absolute constant $c>0$ such that every connected vertex-transitive graph $G$ on $n \ge 3$ vertices contains a cycle of length at least $cn$. Moreover, every such graph with sufficiently large degree $d$ contains a cycle of length at least $(1-d^{-1/100})n$. This gives the first linear bound t...
Let $\tau(G)$ denote the maximum number of pairwise edge-disjoint spanning trees contained in a graph $G$. Zhang and Fan [J. Graph Theory, 2026] posed the problem of determining the corresponding tight spectral-radius condition for every $s$-edge-connected graph with \[ k+1\leq s\leq2k-1, \] together with a characteriz...
Jia Wei, Mu-Huo Liu, Zhi-Fu You et al.· 0 citations
An antimagic labelling of a graph G = (V,E) is a bijection from E to {1,2,...,|E|}, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A graph is antimagic if it has an antimagic labelling. Hartsfield and Ringel in 1990 stated th...
Gr'egoire Beaudoire, C. Bentz, C. Picouleau· 0 citations
Given an integer $k$, deciding whether a graph has a clique of size $k$ is an NP-complete problem. Wilf's inequality provides a spectral lower bound for the clique number of a graph. Strengthening this bound, Elphick and Wocjan proposed a conjecture in 2018 using positive adjacency eigenvalues. In this paper, we presen...
Hareshkumar Jadav, Sudipta Mallik, Akhilesh Panchangam et al.· 0 citations
For a subgraph of a vertex set of a graph to be strong efficient, all remaining vertices in the graph must have exactly one neighbor of equal or greater degree in the specified subgraph and all vertices in the specified subgraph are independent of each other. The strong efficient domination number is the cardinality of...
Betul Atay· Communications faculty of sc...· 0 citations
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