  We study the LHC phenomenology of flavor changing Yukawa couplings between
top quark, Higgs boson, and either an up or charm quark. Such $tuh$ or $tch$
couplings arise for instance in models in which Higgs sector is extended by the
existence of additional Higgs bosons or by higher dimensional operators. We
emphasize the importance of anomalous single top plus Higgs production in these
scenarios, in addition to the more widely studied $t \to h j$ decays. By
recasting existing CMS searches in multilepton and diphoton plus lepton final
states, we show that bounds on $tuh$ couplings are improved by a factor of 1.5
when single top plus Higgs production is accounted for. We also recast the CMS
search for vector boson plus Higgs production into new, competitive constraints
on $tuh$ and $tch$ couplings, setting the limits of $BR(t\to hu) < 0.7%$ and
$BR(t\to hc) <1.2%$. We then investigate the sensitivity of future searches in
multilepton channel and in fully hadronic channel. In multilepton searches,
studying the lepton rapidity distributions and charge assignments can be used
to discriminate between $tuh$ couplings, for which anomalous single top
production is relevant, and $tch$ couplings, for which it is suppressed by the
parton distribution function of the charm quark. An analysis of fully hadronic
$t+h$ production and $t\to h j$ decay can be competitive with the multilepton
search at 100 fb$^{-1}$ of 13 TeV data if jet substructure techniques are
employed to reconstruct boosted top quarks and Higgs bosons. To show this we
develop a modified version of the HEPTopTagger algorithm, optimized for tagging
$t \to h j$ decays. Our sensitivity estimates on $BR(t\to hu)$ ($BR(t\to hc)$)
at 100 fb$^{-1}$ of 13 TeV data for multilepton searches, vector boson plus
Higgs search and fully hadronic search are $0.22%$ ($0.33%$), $0.15%$ ($0.19%$)
and $0.36%$ ($0.48%$), respectively.
