Pretok celih števil skozi točkovno-tranzitivne grafe: Študija simetrije v grafih / Integer flow through vertex-transitive graphs: A study of symmetry in graphs

Raziskovanje simetrije grafov je pomembno v teoriji grafov in vpliva na mnoga področja. Z razumevanjem simetrije lahko bolje spoznamo strukturegrafov. Grafi z visoko stopnjo simetrije, še posebej tisti, ki so tockovno-tranzitivni (npr. Cayleyjevi grafi), običajno kažejo redne vzorce in ponavljajočese strukture, kar omogoča reševanje problemov, povezanih s teorijo grafov. Ta predlog si ambiciozno prizadeva ne le raziskovati strukturo inparametre grafov, ki kažejo določeno stopnjo simetrije, ampak tudi pionirati inovativne metodologije in vpoglede, ki bi lahko preoblikovali našerazumevanje in uporabo grafovskih simetriji. Ocenili bomo veljavnost več znanih domnev, povezanih s simetričnimi grafi. Na primer, ena od znanihdomnev glede grafov s simetrijo, pogosto obravnavana kot najtežja pri problemih simetrije, trdi, da vsak Cayleyjev graf vsebuje Hamiltonovo pot,trditev, ki ostaja nedokazana do danes. Eden od naših ciljev je raziskati to domnevo preko analize koncepta, imenovanega pretoki. Nato se trudimouporabiti snarke kot strategični pristop k reševanju domneve. Ta pristop ne le ponuja pot k raziskovanju domneve, ampak tudi uvaja strukturiranokvir.

The exploration of graph symmetry is important in graph theory and affects many areas. By understanding the symmetries, we can learn more aboutthe graph structures. Graphs with a high degree of symmetry, especially those like vertex-transitive graphs (e.g., Cayley graphs), typically displayregular patterns and recurrent structures, presenting opportunities for problem-solving strategies related to graph theory. This proposal ambitiouslyseeks not only to explore the structure and parameters of graphs exhibiting a certain degree of symmetry but also to pioneer innovativemethodologies and insights that could redefine our understanding and application of graph symmetries. We will assess the validity of several well-known conjectures related to symmetric graphs. For instance, one renowned conjecture regarding graphs with symmetry, often considered the mostchallenging in symmetry problems, asserts that every Cayley graph contains a Hamiltonian cycle, an assertion that remains unproven to date. One ofour objectives is to explore this conjecture through the analysis of a concept known as flows. Subsequently, we endeavor to employ snarks as astrategic approach to address the conjecture. This approach not only offers a pathway to explore the conjecture but also introduces a structuredframework.