Preferred IUPAC names for branched unsaturated hydrocarbons (polyenes)?
According to the general methodology described in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), a double bond is expressed by changing the ending ‘ane’ of the name of the corresponding saturated parent structure to ‘ene’.


  P-31.1.1.1 The presence of one or more double or triple bonds in an otherwise saturated parent hydride (…) is denoted by changing the ending ‘ane’ of the name of a saturated parent hydride to ‘ene’ or ‘yne’. Locants as low as possible are given to multiple bonds as a set, even though this may at times give ‘yne’ endings lower locants than ‘ene’ endings. If a choice remains, preference for low locants is given to the double bonds. In names, the ending ‘ene’ always precedes ‘yne’, with elision of the final letter ‘e’ in ‘ene’. Only the lower locant for a multiple bond is cited, except when the numerical difference between the two locants is greater than one, in which case the higher locant is enclosed in parentheses.
  
  (…)


For example:



If the compound contains multiple double bonds, a multiplicative prefix is placed before the ending ‘ene’ and the letter ‘a’ is inserted between the root of the name of the parent structure and the multiplicative prefix.


  P-31.1.1.2 The multiplying prefixes ‘di’, ‘tri’, etc., are placed before endings denoting unsaturation to indicate the number of multiple bonds of each kind, as required, for example, ‘diene’ and ‘triyne’. For euphonic reasons, when the endings ‘ene’ and ‘yne’ are preceded by a multiplying prefix and a locant the letter ‘a’ is inserted. There is no elision of the final letter ‘a’ of a multiplying prefix before ‘ene’ or ‘yne’, for example, ‘tetraene’ and ‘pentayne’.


For example:



The names of substituent groups are derived from the respective parent hydrides. The presence of free valences formally derived from the loss of one, two, or three hydrogen atoms at the end of the longest chain of a parent hydride is denoted by replacing the ending ‘ane’ of the parent hydride name by ‘yl’, ‘ylidene’, and ‘ylidyne’, respectively.


  The suffixes ‘yl’, ‘ylidene’, and ‘ylidyne’ replace the ending ‘ane’ of the parent hydride name. The atom with the free valence terminates a chain and always has the locant ‘1’, which is omitted from the name. This method is recommended primarily for saturated acyclic and monocyclic hydrocarbon substituent groups (…). Substituent groups formed by this method are referred to as ‘alkyl-type substituent groups’;


For example:


methyl $(\ce{-CH3})$
methylidene $(\ce{=CH2})$
ethyl $(\ce{-CH2-CH3})$
ethenyl $(\ce{-CH=CH2})$
ethylidene $(\ce{=CH-CH3})$


Therefore, the name for the first compound given in the question is ‘3-ethenylhexa-1,5-diene’.



The name for the second compound is ‘3-methylidenehexa-1,5-diene’.



For the third compound, there is a choice among two different principal chains. In accordaqnce with the above-mentioned rules, the name could be either ‘4-ethenylhexa-1,4-diene’ or ‘3-ethylidenehexa-1,5-diene’.

The current IUPAC Recommendations stipulate various criteria for the selection of the principal chain. The first relevant criterion in this case provides that the principal chain has the greater number of skeletal atoms; i.e., the principal chain is the longest chain. If this criterion does not effect a choice of a senior parent structure, the following relevant criteria are applied successively until there are no alternatives remaining. According to Subsection P-44.4.1, the principal chain


  (a) has the greater number of multiple bonds (P-44.4.1.1);
  
  (b) has the greater number of double bonds (P-44.4.1.2);
  
  (…)
  
  (j) has the lower locant(s) for endings or prefixes that express changes in the level of hydrogenation, i.e., for ‘ene’ and ‘yne’ endings and ‘hydro/dehydro’ prefixes (P-44.4.1.10);
  
  (…)


Therefore, the correct name for the third compound is ‘4-ethenylhexa-1,4-diene’ (not ‘3-ethylidenehexa-1,5-diene’) since the locant set ‘1,4’ for the ‘ene’ ending is lower than ‘1,5’.