{"paper":{"title":"ON $(\\triangle, 1)$-GRAPHS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Kelmans, Rafael Aparicio","submitted_at":"2018-06-16T23:58:47Z","abstract_excerpt":"Let $G = (V, E)$ be a graph and $\\lambda $ a non-negative integer. A graph $G$ is called a $(\\lambda, 1)$-{\\em graph} if $ (c0)$ $G$ is neither a complete graph no an edge-empty graph, $ (c1)$ every edge in $G$ belongs to exactly $\\lambda$ triangles, and $(c2)$ every two non-adjacent vertices in $G$ are the end-vertices of exactly one two-edge path in $G$. It turns out that there are infinitely many feasible 4-tuples $(v, d, \\lambda, 1)$ with $\\lambda \\ge 1$. On the other hand (and this is our main result), there is no $(v, d, \\lambda, 1)$-graphs with $\\lambda \\ge 1$. As a byproduct, we obtain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06315","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}