Carl Johan Casselgren

On interval edge colorings of (α,β)-biregular bipartite graphs

Abstract A bipartite graph G is called ( α , β ) -biregular if all vertices in one part of G have degree α and all vertices in the other part have degree β . An edge coloring of a graph G with colors 1 , 2 , 3 , … , t is called an interval t-coloring if the colors received by the edges incident with each vertex of G are distinct and form an interval of integers and at least one edge of G is colored i, for i = 1 , … , t . We show that the problem to determine whether an ( α , β ) -biregular bipartite graph G has an interval t-coloring is NP -complete in the case when α = 6 , β = 3 and t = 6 . It is known that if an ( α , β ) -biregular bipartite graph G on m vertices has an interval t-coloring then α + β - gcd ( α , β ) ⩽ t ⩽ m - 1 , where gcd ( α , β ) is the greatest common divisor of α and β . We prove that if an ( α , β ) -biregular bipartite graph has m ⩾ 2 ( α + β ) vertices then the upper bound can be improved to m - 3 .

Avoiding Arrays of Odd Order by Latin Squares

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each s ...

On Path Factors of (3, 4)-Biregular Bigraphs

A (3, 4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.

Completing partial Latin squares with one filled row, column and symbol

Let P be an n x n partial Latin square every non-empty cell of which lies in a fixed row r, a fixed column c or contains a fixed symbols. Assume further that s is the symbol of cell (r, c) in P. We ...

Coloring graphs from random lists of size 2

Let G=G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant @D. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size @s(n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring @f such that @f(v)@?L(v) for all v@?V(G). In particular, we show that if g is odd and @s(n)=@w(n^1^/^(^2^g^-^2^)), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n->~. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n>=g, there is a graph H=H(n,g) with bounded maximum degree and girth g, such that if @s(n)=o(n^1^/^(^2^g^-^2^)), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n->~. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size @s(n), exhibits a sharp threshold at @s(n)=2n.

Some results on cyclic interval edge colorings of graphs

A proper edge coloring of a graph

Coloring graphs from random lists of fixed size

G

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

with colors

Coloring graphs of various maximum degree from random lists

1,2,\dots,t

Solution of Vizing's Problem on Interchanges for the case of Graphs with Maximum Degree 4 and Related Results

is called a \emph{cyclic interval