For loopless multigraphs \(G\), the total choice number is asymptotically equal to its fractional counterpart as the latter invariant tends to infinity. If \(G\) is embedded in the plane, then the edge-face and entire choice numbers exhibit the same “asymptotically good” behaviour. These results are based mainly on an analogous theorem of Kahn [5] for the list-chromatic index. Together with work of Kahn and others, our three results give a complete answer to a natural question: which of the seven invariants associated with list-colouring the nonempty subsets of \(\{V, E, F\}\) are asymptotically good?

In 1970, Behzad, Chartrand and Wall conjectured that the girth of every \(r\)-regular digraph \(G\) of order \(n\) is at most \(\left\lceil \frac{n}{r} \right\rceil\). The conjecture follows from a theorem of Menger and Dirac if \(G\) has strong connectivity \(x = r\). We show that any digraph with minimum in-degree and out-degree at least \(r\) has girth at most \(\left\lceil \frac{n}{r} \right\rceil\) if \(\kappa = r – 1\). We also find from the literature a family of counterexamples to a conjecture of Seymour.

In this paper, we give an alternative proof for the fact that the graph obtained by overlapping the cycle \(C_m\) (\(m \geq 3\)) and the complete bipartite graph \(K_{2,s}\) (\(s \geq 1\)) at an edge is uniquely determined by its chromatic polynomial. This result provides a partial solution to a question raised in [7].

Let \(G\) be a simple graph with \(n\) vertices. \(p(G,k)\) denotes the number of ways in which one can select \(k\) independent edges in \(G\) (\(k \geq 1\)). Let \(p(G,0) = 1\) for all \(G\).
The matching polynomial \(\alpha(G)\) of a graph \(G\) is given by:
\[\alpha(G) = \alpha(G,x) = \sum_{k=0}^{\left[\frac{n}{2}\right]} (-1)^k p(G,k) x^{n-2k}\]

In this article, we give the matching polynomials of the complete \(n\)-partite graph with a differential operator.

The List Edge Coloring Conjecture states that for every graph, the chromatic index equals the choice index. We prove the conjecture for outerplanar graphs with maximum degree at least five.

Cycle prefix digraphs are a class of Cayley coset graphs with many remarkable properties, such as:Symmetry Large number of nodes for a given degree and diameter Simple shortest path routing Hamiltonicity Optimal connectivity Others.
In this paper, we show that the cycle prefix digraphs, like the Kautz digraphs, contain cycles of all lengths \(l\), with \(l\) between two and \(N\), the order of the digraph, except for \(N-1\).

Let \(G\) be a cubic bipartite plane graph that has a perfect matching. If \(M\) is any perfect matching of \(G\), then \(G\) has a face that is \(M\)-alternating.If \(f\) is any face of \(G\), then there is a perfect matching \(M\) such that \(f\) is \(M\)-alternating.There is a simple algorithm for visiting all perfect matchings of \(G\) beginning at one.
There are infinitely many cubic plane graphs that have perfect matchings but whose matching transformation graphs are completely disconnected.
Several problems are proposed.

In this paper, we calculate the jump number of the product of an ordered set and a chain.

In [1], [2] we can find results concerning kernel-perfect graphs and solvable graphs. These concepts are related to kernels of a digraph. The authors of [2] consider two graph constructions: the join of two graphs and duplication of a vertex. These kinds of graphs preserve kernel-perfectness and solvability of their orientations. In this paper we generalize results from [2] applying them to \((k,l)\)-kernels and two operations: generalized join and duplication of a subset of vertices. The concept of a \((k,l)\)-kernel of a digraph was introduced in [8] and was studied in [6], [7], and [9]. In our considerations we take advantage of the asymmetrical part of digraphs, which was used by H. Galeana-Sanchez in [6] in the proof of a sufficient condition for a digraph to have a \((k, l)\)-kernel.

With the help of computer algorithms, we improve the lower bound on the Ramsey multiplicity of \(K_4\) and thus show that the exact value of it is equal to \(9\).