Conjecture Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .
An alternating walk in a digraph is a walk so that the vertex is either the head of both and or the tail of both and for every . A digraph is universal if for every pair of edges , there is an alternating walk containing both and
Question Does there exist a locally finite highly arc transitive digraph which is universal?
Conjecture For every , the sequence in consisting of copes of and copies of has the fewest number of distinct subsequence sums over all zero-free sequences from of length .
Conjecture If is a bridgelesscubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .
Conjecture There is an integer-valued function such that if is any -connected graph and and are any two vertices of , then there exists an induced path with ends and such that is -connected.
The deck of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).
Conjecture If two graphs on vertices have the same deck, then they are isomorphic.
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be a graph and let 
such that for any pair 
there are 
edge-disjoint paths from 
to 
in 
edge-disjoint trees, each of which contains 
. 
such that for every hexagonal graph 
, 
, what is the smallest integer 
such that 
? 
so that the vertex 
is either the head of both 
and 
or the tail of both 
. A digraph is universal if for every pair of edges 
, there is an alternating walk containing both 
and 
, the sequence in 
consisting of 
copes of 
and 
copies of 
has the fewest number of distinct subsequence sums over all zero-free sequences from 
. 
is the only 
-chromatic double-critical graph 
be a prime natural number. Find all primes 
, such that 
.
for every endo-![\[ M_n = 2^p - 1 \]](/files/tex/eb18a56e5c2e8b1be6ac733d217c0c1f1a47a94e.png)
that is prime.
of 
has equivalence covering number 
. 
-degenerate graph (a forest) and a 
-degenerate graph is 
-colourable. 
such that if 
and 
are any two vertices of 
with ends 
is 
concentrated on a single value? 
and graph 
. 
immerses 
. 
and 
are 
, then 
has distinct subset sums if distinct subsets of 
have distinct sums.
whenever 
has distinct subset sums. 
of 
. 
vertices have the same deck, then they are isomorphic. 
is unfriendly if every vertex has at least as many neighbors in the other classes as in its own. 
contains every oriented tree of order 
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