## Graduate Theses, Dissertations, and Problem Reports

### Edge colorings of graphs on surfaces and star edge colorings of sparse graphs

Spring

2019

Dissertation

PhD

#### College

Eberly College of Arts and Sciences

Mathematics

Rong Luo

John Goldwasser

John Goldwasser

Hong-Jian Lai

Cun-Quan Zhang

Yue Zhao

#### Abstract

In my dissertation, I present results on two types of edge coloring problems for graphs.

For each surface Σ, we define ∆(Σ) = max{∆(G)| G is a class two graph with maximum degree ∆(G) that can be embedded in Σ}. Hence Vizing’s Planar Graph Conjecture can be restated as ∆(Σ) = 5 if Σ is a sphere. For a surface Σ with characteristic χ(Σ) ≤ 0, it is known ∆(Σ) ≥ H(χ(Σ))−1, where H(χ(Σ)) is the Heawood number of the surface, and if the Euler char- acteristic χ(Σ) ∈ {−7, −6, . . . , −1, 0}, ∆(Σ) is already known. I study critical graphs on general surfaces and show that (1) if G is a critical graph embeddable on a surface Σ with Euler character- istic χ(Σ) ∈ {−6, −7}, then ∆(Σ) = 10, and (2) if G is a critical graph embeddable on a surface Σ with Euler characteristic χ(Σ) ≤ −8, then ∆(G) ≤ H(χ(Σ)) (or H(χ(Σ))+1) for some special families of graphs, namely if the minimum degree is at most 11 or if ∆ is very large et al. As applications, we show that ∆(Σ) ≤ H (χ(Σ)) if χ(Σ) ∈ {−22, −21, −20, −18, −17, −15, . . . , −8}and ∆(Σ) ≤ H (χ(Σ)) + 1 if χ(Σ) ∈ {−53, . . . , 23, −19, −16}. Combining this with , it follows that if χ(Σ) = −12 and Σ is orientable, then ∆(Σ) = H(χ(Σ)).

A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored sub-

graph is a path of length at most 3. The star chromatic index χ′st(G) of a graph G is the smallest

integer k such that G has a star k-edge-coloring. The list star chromatic index ch′st(G) is defined

analogously. Bezegova et al. and Deng et al. independently proved that χ′ (T) ≤ 3∆ for anyst 2

tree T with maximum degree ∆. Here, we study the list star edge coloring and give tree-like

bounds for (list) star chromatic index of sparse graphs. We show that if mad(G) < 2.4, then

χ′ (G)≤3∆+2andifmad(G)<15,thench′ (G)≤3∆+1.Wealsoshowthatforeveryε>0st 2 7 st 2

there exists a constant c(ε) such that if mad(G) < 8 − ε, then ch′ (G) ≤ 3∆ + c(ε). We also3 st 2

find guaranteed substructures of graph with mad(G) < 3∆ − ε which may be of interest in other2

problems for sparse graphs.

COinS

#### DOI

https://doi.org/10.33915/etd.3910