
Extending Edgecolorings of Complete Hypergraphs into Regular Colorings
Let Xh be the collection of all hsubsets of an nset X⊇ Y. Given a colo...
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Graph multicoloring reduction methods and application to McDiarmidReed's Conjecture
A (a,b)coloring of a graph G associates to each vertex a set of b color...
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On the balanceability of some graph classes
Given a graph G, a 2coloring of the edges of K_n is said to contain a b...
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3Coloring on Regular, Planar, and Ordered Hamiltonian Graphs
We prove that 3Coloring remains NPhard on 4 and 5regular planar Hami...
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On completely regular codes of covering radius 1 in the halved hypercubes
We consider constructions of coveringradius1 completely regular codes,...
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Regular subgroups with large intersection
In this paper we study the relationships between the elementary abelian ...
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Extremal values of semiregular continuants and codings of interval exchange transformations
Given a set A consisting of positive integers a_1<a_2<⋯<a_k and a kterm...
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On Regular Set Systems Containing Regular Subsystems
Let X,Y be finite sets, r,s,h, λ∈ℕ with s≥ r, X⊊ Y. By λXh we mean the collection of all hsubsets of X where each subset occurs λ times. A coloring of λXh is rregular if in every color class each element of X occurs r times. A oneregular color class is a perfect matching. We are interested in the necessary and sufficient conditions under which an rregular coloring of λXh can be embedded into an sregular coloring of λYh. Using algebraic techniques involving glueing together orbits of a suitably chosen cyclic group, the first author and Newman (Combinatorica 38 (2018), no. 6, 1309–1335) solved the case when λ=1,r=s, (X,Y,h)=(Y,h). Using purely combinatorial techniques, we nearly settle the case h=4. Two major challenges include finding all the necessary conditions, and obtaining the exact bound for Y. It is worth noting that completing partial symmetric latin squares is closely related to the case λ =r=s=1, h=2 which was solved by Cruse (J. Comb. Theory Ser. A 16 (1974), 18–22).
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