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Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs
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Proper path-factors and interval edge-coloring of
(3, 4)-biregular bigraphs
Armen S. Asratian?, Carl Johan Casselgren?,
Jennifer Vandenbussche?, Douglas B. West§
April 6, 2007
Abstract
An interval coloring of a graph G is a proper coloring of E(G) by positive integers such
that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph
is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the
other has degree 4; it is unknown whether these all have interval colorings. We prove that G
has an interval coloring using 6 colors when G is a (3, 4)-biregular bigraph having a spanning
subgraph whose components are paths with endpoints at 3-valent vertices and lengths in
{2, 4, 6, 8}. We provide sufficient conditions for the existence of such a subgraph.
Keywords: path factor, interval edge-coloring, biregular bipartite graph
AMSclass: 05C15, 05C70
1 Introduction
An interval coloring or consecutive coloring of a graph G is a proper coloring of the edges of G
by positive integers such that the colors on the edges incident to any vertex are consecutive. The
notion was introduced by Asratian and Kamalian [2] (available in English as [3]), motivated by the
problem of constructing timetables without “gaps” for teachers and classes. Hansen [9] suggested
another scenario: a school wishes to schedule parent-teacher conferences in time slots so that every
person’s conferences occur in consecutive slots. A solution exists if and only if the bipartite graph
with vertices for the people and edges for the required meetings has an interval coloring.
?Linko?ping University, Linko?ping Sweden, arasr@mai.liu.se.
?Ume?a University, Ume?a , Sweden, carl-johan.casselgren@math.umu.se.
?University of Illinois, Urbana, IL, jarobin1@math.uiuc.edu.
§University of Illinois, Urbana, IL, west@math.uiuc.edu. Work supported in part by the NSA under Award
No. H98230-0
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