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...
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...
Maxwell observed that the graph of any rigid generic framework in $\mathbb{R}^d$ on $n$ vertices has at least $dn-\binom{d+1}{2}$ edges. In this article we prove that graphs whose complement has maximum degree at most two and no component isomorphic to a triangle or a square are rigid in the maximum dimension allowed b...
For a finite simple graph G, let alpha(G) denote its independence number and let l_avg(G) = (1 / |V(G)|) sum_{v in V(G)} alpha(G[N_G(v)]) be the average independence number of its open neighbourhoods. Written on the Wall II Conjecture 194 asserts that every simple connected graph on n>1 vertices satisfying alpha(G)<= 1...
In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters d1≤d2, there exists a coupling of G(n,d1) and G(n,d2) satisfying that G(n,d1)⊆G(n,d2) with high probability, confirming a conjecture of Ga...
Let $G$ be a connected graph of order $n$. Gallai conjectured that $G$ can be decomposed into $\frac{n+1}{2}$ paths. This is known as Gallai’s conjecture of path decomposition. In this paper, we focus on path decomposition of product graphs. It is proved that $G□H$ can be decomposed into $\frac{n(G□H)}{2}$ paths if $G$...
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