After the break up of Austro-Hungarian empire, State of Slovenes, Croats and Serbs (that were living inside A-H) was formed on 29.10.1918 and merged with Kingdom of Serbia on 1.12.1918, that was then renamed Kingdom of Serbs, Croats and Slovenes. First serbian (as from actual Serbia) battalion came to "yugoslav"-austrian border on 23.12.1918.
The border between Austrian and Slovene nationalities was not the same as the border between the regions, so, naturally, a "small war" broke out. It all started when (first slovene if anyone cares

) general, Rudolf Maister and his slovene disarmed austrian militia in town of Maribor (region Styria) just a day before austrian militia planned to disarm them. That happened on 1.11.1918. They kept advancing and have liberated (or occupied, depends on the perspective) the area that is now a northern border between Slovenia and Austria. Other slovene units were advancing in Corinthia. Corinthia lies high in Alpine region and a mountain pass (that is today a border between the two countries) was occupied by Slovene forces in 19.11.1918.
At the end of november 1918, Slovenes had 1,600 men with 10 guns and Austrians had around 3,400 men and 7 guns in Corinthia.
On 1.12.1918 Austrians launched a counter offensive and they scored some success. By the end of the year, Slovene forces got some reinforcements and were at that time 2,900 men strong and had 12 guns.
On 7.1.1919, Slovene forces were pushed almost across the Alps. On 13.1. hostilities ended. Cease fire came to effect on 19.1. as they were waiting for an american colonel (later general) Sherman Miles.
The two armies then bombarded each other occasionally, but didn't cross the decided lines until 29.4.1919, when Slovenes launched an offensive on whole lenght of the Corinthia. As the attack was ill prepared, they've gained some ground at start and then fallen back to the starting positions.
Austrians counter attacked on 2.5.1919, routed the Slovenes and kept pushing. There were some cease fire negotiations started on 9.5. but weren't going anywhere so the talks ceased on 16.5.
In the mean while, general mobilisation was started in Slovene area on 3.5. Yugoslav (well, technically Kingdom of SHS) offensive was launched on 28.5. Yugoslav army had 22 battalions, which was roughly 12,000 men. Austrians had around 11,000 men in militia and military units. There were some talks about cease fire, but Yugoslav army kept pushing until it reached Klagenfurt on 6.6.1919. In the mean time, it was decided that a plebiscite would be held to decide who gets the Corinthia. It was held on 20.10.1920 and people have voted to join Austria.
To answer the question, Slovene soldiers that were fighting with Austrian soldier had uniforms and equipment from Austro-Hungarian empire. The only thing different were cockades and hats, that were a completely new design. But I've read in another source that Slovenes had to give up their hats as they've apparently took over serbian uniforms by June/July 1919, when Slovene, Croat and forces from Kingdom of Serbia were formed into one army. That time was also the end of slovene language being used as a commanding language in Slovene units and officers were mixed a bit.
Kingdom of SHS was formally renamed into Kingdom of Yugoslavia in 1929, but the term was used before that colloquially.
Here are some photos:
"Slovene soldiers in trenches in Corinthia, early in 1919"
http://www.hervardi.com/images/vojaki_koroska_1919.jpgBattle for Gornja Radgona:
http://www.hervardi.com/images/bitka_za_radgono004.jpgGeneral Maister on a visit (those are the new hats they are wearing, I think only officers and NCOs got them):
http://www.hervardi.com/images/bitka_za_radgono001.jpgMaister's officers (mind the hat):
http://upload.wikimedia.org/wikipedia/commons/b/b9/Maistrovi_borci.jpgSoldiers that fought around Črna na Koroškem (Corinthia):
http://www.hervardi.com/images/borci_celjske_legije.jpgA group photo of Slovenes. You can see some of the "slovene" hats and others appear to be of a Serbian design:
http://www.hervardi.com/images/tabor_mp007.jpg