Strong parity edge-colorings of graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915024368304128 |
|---|---|
| author | Bradshaw, Peter Norin, Sergey West, Douglas B. |
| author_facet | Bradshaw, Peter Norin, Sergey West, Douglas B. |
| contents | An edge-coloring of a graph $G$ assigns a color to each edge of $G$. An edge-coloring is a parity edge-coloring if for each path $P$ in $G$, it uses some color on an odd number of edges in $P$. It is a strong parity edge-coloring if for every open walk $W$ in $G$, it uses some color an odd number of times along $W$. The minimum numbers of colors in parity and strong parity edge-colorings of $G$ are denoted $p(G)$ and $\hat{p}(G)$, respectively.
We characterize strong parity edge-colorings and use this characterization to prove lower bounds on $\hat{p}(G)$ and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that $\hat{p}(K_{s,t})=s \circ t$, where $s \circ t$ is the Hopf-Stiefel function. (2) We show that $\hat{p}(G)$ for a connected $n$-vertex graph $G$ equals the known lower bound $\lceil \log_2 n \rceil$ if and only if $G$ is a subgraph of the hypercube $Q_{\lceil \log_2 n \rceil }$. (3) We asymptotically compute $\hat{p}(G)$ when $G$ is the $\ell$th distance-power of a path, proving $\hat{p}(P_n^\ell)\sim\ell \lceil {\log_2 n} \rceil$. (4) We disprove the conjecture that $\hat{p}(G)=p(G)$ when $G$ is bipartite by constructing bipartite graphs $G$ such that $\hat{p}(G)/p(G)$ is arbitrarily large; in particular, with $\hat{p}(G)\ge\frac{1-o(1)}3 k\ln k$ and $p(G)\le2k+k^{1/3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong parity edge-colorings of graphs Bradshaw, Peter Norin, Sergey West, Douglas B. Combinatorics 05C15 An edge-coloring of a graph $G$ assigns a color to each edge of $G$. An edge-coloring is a parity edge-coloring if for each path $P$ in $G$, it uses some color on an odd number of edges in $P$. It is a strong parity edge-coloring if for every open walk $W$ in $G$, it uses some color an odd number of times along $W$. The minimum numbers of colors in parity and strong parity edge-colorings of $G$ are denoted $p(G)$ and $\hat{p}(G)$, respectively. We characterize strong parity edge-colorings and use this characterization to prove lower bounds on $\hat{p}(G)$ and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that $\hat{p}(K_{s,t})=s \circ t$, where $s \circ t$ is the Hopf-Stiefel function. (2) We show that $\hat{p}(G)$ for a connected $n$-vertex graph $G$ equals the known lower bound $\lceil \log_2 n \rceil$ if and only if $G$ is a subgraph of the hypercube $Q_{\lceil \log_2 n \rceil }$. (3) We asymptotically compute $\hat{p}(G)$ when $G$ is the $\ell$th distance-power of a path, proving $\hat{p}(P_n^\ell)\sim\ell \lceil {\log_2 n} \rceil$. (4) We disprove the conjecture that $\hat{p}(G)=p(G)$ when $G$ is bipartite by constructing bipartite graphs $G$ such that $\hat{p}(G)/p(G)$ is arbitrarily large; in particular, with $\hat{p}(G)\ge\frac{1-o(1)}3 k\ln k$ and $p(G)\le2k+k^{1/3}$. |
| title | Strong parity edge-colorings of graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2411.11124 |