Let G be a nonempty finite simple graph of order n, and let m(G) be the upper median of its degree sequence. We prove that the 2-domination number satisfies gamma_2(G)<= n - m(G) + 1. This proves Graffiti.pc Conjecture 387. In fact, the argument establishes the inequality for every nonempty finite simple graph, so the connectedness hypothesis in the original formulation is unnecessary. The proof uses the complement graph and a minimally linearly dependent family of polynomials encoding selected nonneighborhoods.
Albertson's conjecture states that every graph of chromatic number r has crossing number at least that of the complete graph K_r. We prove the conjecture for all r. The proof combines new clique-sensitive edge bounds for critical graphs, maximum-clique sampling, and a reserved immersion construction. For graphs of orde...
We prove the conjecture of Xu, Li, Hua, and Pan that the resistance diameter does not increase under the line-graph operation. For every finite connected simple graph $G$ with at least one edge, we establish $D_r(L(G))\le D_r(G)$, with equality if and only if $G$ is a cycle or $K_4$. The proof combines an exact electri...
In 1971, Mason conjectured that the numbers of independent sets of fixed size in a matroid constitute an ultra log-concave sequence. In 2020, this conjecture was proven by Br\"and\'en and Huh and independently by Anari, Liu, Gharan and Vinzant. Recently, this result was extended to $M^\natural$-concave functions. In th...
Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [...
A graph is
perfect
if, for every induced subgraph, the chromatic number equals the size of its largest clique. In 1972, Lovász established a fundamental characterization of perfect graphs, showing that a graph is perfect if and only if, for every induced subgraph, the product of the size of the largest independen...
K\"{u}hn, Sauermann, Steiner and Wigderson recently disproved the Odd Hadwiger Conjecture, even for graphs with independence number 2. For this class of graphs the conjecture is known to be equivalent to the following: every $n$-vertex graph $G$ with independence number 2 contains $K_{\lceil \frac n2 \rceil}$ as an odd...
Antonia Bermúdez, Bruno L. Netto, Daniel A. Quiroz· 0 citations
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