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1
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In-degree sequence in a general model of a random digraph

100%
Palka Z., Sperling M.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
193-207
EN
A general model of a random digraph D(n,P) is considered. Based on a precise estimate of the asymptotic behaviour of the distribution function of the binomial law, a problem of the distribution of extreme in-degrees of D(n,P) is discussed.
2
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Degree sequences of digraphs with highly irregular property

80%
Majcher Z., Michael J.
Discussiones Mathematicae Graph Theory
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1998
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tom 18
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nr 1
49-61
EN
A digraph such that for each its vertex, vertices of the out-neighbourhood have different in-degrees and vertices of the in-neighbourhood have different out-degrees, will be called an HI-digraph. In this paper, we give a characterization of sequences of pairs of out- and in-degrees of HI-digraphs.
3
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A Constructive Extension of the Characterization on PotentiallyK s , t -Bigraphic Pairs

80%
Guo J.-Y., Yin J.-H.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
251-259
EN
Let Ks,t be the complete bipartite graph with partite sets of size s and t. Let L1 = ([a1, b1], . . . , [am, bm]) and L2 = ([c1, d1], . . . , [cn, dn]) be two sequences of intervals consisting of nonnegative integers with a1 ≥ a2 ≥ . . . ≥ am and c1 ≥ c2 ≥ . . . ≥ cn. We say that L = (L1; L2) is potentially Ks,t (resp. As,t)-bigraphic if there is a simple bipartite graph G with partite sets X = {x1, . . . , xm} and Y = {y1, . . . , yn} such that ai ≤ dG(xi) ≤ bi for 1 ≤ i ≤ m, ci ≤ dG(yi) ≤ di for 1 ≤ i ≤ n and G contains Ks,t as a subgraph (resp. the induced subgraph of {x1, . . . , xs, y1, . . . , yt} in G is a Ks,t). In this paper, we give a characterization of L that is potentially As,t-bigraphic. As a corollary, we also obtain a characterization of L that is potentially Ks,t-bigraphic if b1 ≥ b2 ≥ . . . ≥ bm and d1 ≥ d2 ≥ . . . ≥ dn. This is a constructive extension of the characterization on potentially Ks,t-bigraphic pairs due to Yin and Huang (Discrete Math. 312 (2012) 1241–1243).
4
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Degree Sequences of Monocore Graphs

80%
Bickle A.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 3
585-592
EN
A k-monocore graph is a graph which has its minimum degree and degeneracy both equal to k. Integer sequences that can be the degree sequence of some k-monocore graph are characterized as follows. A nonincreasing sequence of integers d0, . . . , dn is the degree sequence of some k-monocore graph G, 0 ≤ k ≤ n − 1, if and only if k ≤ di ≤ min {n − 1, k + n − i} and ⨊di = 2m, where m satisfies [...] ≤ m ≤ k ・ n − [...] .
5
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3-Paths in Graphs with Bounded Average Degree

80%
Jendrol′ S., Maceková M., Montassier M., Soták R.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 2
339-353
EN
In this paper we study the existence of unavoidable paths on three vertices in sparse graphs. A path uvw on three vertices u, v, and w is of type (i, j, k) if the degree of u (respectively v, w) is at most i (respectively j, k). We prove that every graph with minimum degree at least 2 and average degree strictly less than m contains a path of one of the types [...] Moreover, no parameter of this description can be improved.
6
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Solution to the problem of Kubesa

80%
Meszka M.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 2
375-378
EN
An infinite family of T-factorizations of complete graphs $K_{2n}$, where 2n = 56k and k is a positive integer, in which the set of vertices of T can be split into two subsets of the same cardinality such that degree sums of vertices in both subsets are not equal, is presented. The existence of such T-factorizations provides a negative answer to the problem posed by Kubesa.
7
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Potentially H-bigraphic sequences

80%
Ferrara M., Jacobson M., Schmitt J., Siggers M.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 3
583-596
EN
We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y, respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A. Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of S containing H as a subgraph. We define σ(H,m,n) to be the minimum integer k such that every bigraphic pair S = (A,B) with |A| = m, |B| = n and σ(S) ≥ k is potentially H-bigraphic. In this paper, we determine $σ(K_{s,t},m,n)$, σ(Pₜ,m,n) and $σ(C_{2t},m,n)$.
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Structural results on maximal k-degenerate graphs

70%
Bickle A.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 4
659-676
EN
A graph is k-degenerate if its vertices can be successively deleted so that when deleted, each has degree at most k. These graphs were introduced by Lick and White in 1970 and have been studied in several subsequent papers. We present sharp bounds on the diameter of maximal k-degenerate graphs and characterize the extremal graphs for the upper bound. We present a simple characterization of the degree sequences of these graphs and consider related results. Considering edge coloring, we conjecture that a maximal k-degenerate graph is class two if and only if it is overfull, and prove this in some special cases. We present some results on decompositions and arboricity of maximal k-degenerate graphs and provide two characterizations of the subclass of k-trees as maximal k-degenerate graphs. Finally, we define and prove a formula for the Ramsey core numbers.
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