Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects class simplicial created by graph product, which is an extension the pseudo-fractal scale-free web. We determine explicitly independence number, domination chromatic number. Moreover, derive closed-form expr...

A chromatic-index-critical graph G on n vertices is non-trivial if it has at most bn2 c edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial chromaticindex-critical graphs of order 12.

A snark is a cyclically-4-edge-connected cubic graph with chromatic index 4. In 1880, Tait proved that the Four-Color Conjecture is equivalent to the statement that every planar bridgeless cubic graph has chromatic index 3. The search for counter-examples to the FourColor Conjecture motivated the definition of the snarks. A k-total-coloring of G is an assignment of k colors to the edges and ver...

Let G be a simple graph with maximum degree Δ(G). A subgraph H of is overfull if |E(H)|>Δ(G)⌊|V(H)|/2⌋. Chetwynd and Hilton in 1986 conjectured that on n vertices Δ(G)>n/3 has chromatic index Δ(G) only contains no subgraph. Glock, Kühn Osthus 2016 showed the conjecture true for dense quasirandom graphs even order, they same should hold such odd order. In this paper, we show affirmative.

We investigate the effect of a fixed forbidden clique minor upon strong chromatic index, both in multigraphs and simple graphs. conjecture for each k ? 4 that any K -minor-free multigraph maximum degree ? has index at most 3 2 ( ? ) ?. present construction certifying if true is asymptotically sharp as ? ?. In support conjecture, we show it case = prove statement number place index. By contrast,...