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Decomposition of a regular graph and connected subgraphs
Yes, that is right. 
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revised 
Decomposition of a regular graph and connected subgraphs
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Decomposition of a regular graph and connected subgraphs
That would be true if you required your graph to be transitive, but I don't see how mere regularity gets you there. 
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Decomposition of a regular graph and connected subgraphs
The comments on my answer lead me to wonder: is the condition on the successive decompositions supposed to hold for every possible choice of the $x_i$, or just for one? 
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revised 
Decomposition of a regular graph and connected subgraphs
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answered  Decomposition of a regular graph and connected subgraphs 
2d

awarded  Critic 
Jul 27 
awarded  Citizen Patrol 
Dec 2 
comment 
Characterizing orthants with polynomials
It does sound familiar, but all I could find was this question on math.se: math.stackexchange.com/questions/966633/… 
Oct 29 
awarded  Editor 
Oct 29 
revised 
Find all faces in a graph from list of edges
deleted 7 characters in body 
Oct 29 
answered  Find all faces in a graph from list of edges 
Sep 22 
comment 
Characterization of a subset of [0,1] $II$
Given the possibility of excluding arbitrary countable subsets, I guess the restriction has to be that for $t\in T, t\neq 1$, the intersection $T\cap[t,t+\delta]$ must be uncountable for all $\delta>0$. 
Jan 29 
answered  Is there a dense rational sequence of positive separation? 
Jan 16 
comment 
An extension of the real semiring with multiple degrees of infinity
I still don't quite get the "dominance" property of divisibility; assuming $(a/b)*b$ should be $a$, your second example gives $2^*=1^*+2^*$, so that either $1^*=0$, or your semiring is not cancellable  are you ok with the latter? 
Oct 11 
awarded  Caucus 
Oct 11 
awarded  Constituent 
Jun 25 
awarded  Yearling 
May 14 
comment 
two boy scouts problems
Your original problem does sound solvable to me (essentially, you need a $(2n1)$edge coloring of $K_{2n}$). Is there an additional constraint (e.g. only one match of each sport per round)? 
Oct 26 
answered  fixed points of system of quadratic equations 