A note on total and list edge-colouring of graphs of tree-width 3

It is shown that Halin graphs are $\Delta$-edge-choosable and that graphs of tree-width 3 are $(\Delta+1)$-edge-choosable and $(\Delta +2)$-total-colourable.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0212

Total weight choosability in Hypergraphs

A total weighting of the vertices and edges of a hypergraph is called vertex-coloring if the total weights of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this note we show that such a weighting is possible if every vertex has two, and every edge has three weights to choose from, extending a recent result on graphs to hypergraphs

A note on the adjacent vertex distinguishing total chromatic number of graphs

AbstractAn adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have different sets of colors. The minimum number of colors needed for such a total coloring of G is denoted by χa″(G). In this note, we show that χa″(G)≤2Δ for any graph G with maximum degree Δ≥3

Enumeration of graphs with a heavy-tailed degree sequence

In this paper, we asymptotically enumerate graphs with a given degree sequence d=(d_1,...,d_n) satisfying restrictions designed to permit heavy-tailed sequences in the sparse case (i.e. where the average degree is rather small). Our general result requires upper bounds on functions of M_k= \sum_{i=1}^n [d_i]_k for a few small integers k\ge 1. Note that M_1 is simply the total degree of the graphs. As special cases, we asymptotically enumerate graphs with (i) degree sequences satisfying M_2=o(M_1^{ 9/8}); (ii) degree sequences following a power law with parameter gamma>5/2; (iii) power-law degree sequences that mimic independent power-law "degrees" with parameter gamma>1+\sqrt{3}\approx 2.732; (iv) degree sequences following a certain "long-tailed" power law; (v) certain bi-valued sequences. A previous result on sparse graphs by McKay and the second author applies to a wide range of degree sequences but requires Delta =o(M_1^{1/3}), where Delta is the maximum degree. Our new result applies in some cases when Delta is only barely o(M_1^ {3/5}). Case (i) above generalises a result of Janson which requires M_2=O(M_1) (and hence M_1=O(n) and Delta=O(n^{1/2})). Cases (ii) and (iii) provide the first asymptotic enumeration results applicable to degree sequences of real-world networks following a power law, for which it has been empirically observed that 2<gamma<3.Comment: 34 page